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PRINCIPI HIEREDITARIO,
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Burgundiz, Brabantiz , Stiriz, Carinthia; Carniolz,
' Aufirafig , Wirtenbergz &c.. :
Duci POTENTISSIMO;
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tiz , Alfatiz, Illyrici , Vandaliz; Portenavia
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PRINCIPI, COMITI AC DOMINO,
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TETRAGONOMETRIAM
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finé qvimvis, ní ante omnia demonfttéve?i
nummum ;. Hoc eft illud, Da. demon/trandi, hu.
jus perverfi zvi , qvod qvalibet' divin? Mathe:
feas /ícis- Gcometrix "arcana; fujs:culpribur now
tàm parcé, frugaliter & oecenomicé diftribuit, qvàm.
inter hoftes & contemtores copiofisfimé difleminat
& profusé disfipat ; Longé alius antiqvis tempori-
"busfiib Byehagorz, Platone; Archyta & Ariftocele
— Giémetriz fuit cultus; Divinus Plato, Philofophos,
qvidem à.fchola.& otio ad Rempub. gerendam, à
contemplatione artium ad ipfacum, utilitatem. exer-
cendár cocgiffe, A rchyram veró & Eudoxum, qvi
mathetnaticas contemplationes ab animo & rebus
in mentis. inclligentiam tantüm cadentibus, ad re» .
rum fenfilium & cerporearum , exempla traducerg
&: Geomcettiam varietate demonílrarionis non fo--
làm logicz fed etiani practica per inflrumentaria &
machinalia, dryénxd xd) uexeriya €CXOrnare voluerunt,
ex indignatione, qvód nobilisfimam Pbilofophorum -
- voffes&onem in vulgusindicarent ae publicarent, &
velut ascana Philofophiz myfteria proderent, ab in-
fituco dererruiffe, legitur: Sed; hodié minis follicti
furone de.refervasionc ejusmodi Magifleriali, qvic-
| gyamhi rpísyande pro Magiftro, fed pptiüs com-
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imurüter ftudemns, M oikem odis propagare de
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litate & utilitate plenisfimam doctrinam Mathema-
ticatn, qvam neto facile motraliü folus & ex affe ex-
. €olre& minüsadhucad praxin illa omnia deducere
: valebit; qva ingenió affeqvi capax eft, qvaqtopter
. Optandum,ut omnesingenió praditi, tàm nobilisfi:
mo &utilisfimo ftudio ftam havarent operam; Et
profe&ó omnes qvidem naturá 'inclinamus ad pe-
netrandas Mathematicas jucunditatés partim rei
-—
mirabilitate adducti, partim emergenus tandemi:
lucri & commodi gratiá, fed qvàm plurimi diffi-
ciliorem & minüs expectátum labori feritientes
& ad lucrum magis prafens réfpiciéhtes, actutum
refiliunt, ceulevispila, vi ad durum lapidem impul-
fa; Omnium veró primó: permulti :deterrentur a-
. rithmeticz prolixitatibus &' horrendis operationi-
bus &íàné nifi exdem per varia £ólnpendia & ta
bulas paffim elaboratas remotz fuerint, parum aut
ferénihil pratet fpeculationes ad praxin feu ufum de-
duci poffét;Qvapropter elaboratas habdmiis Tabulas
finuum, Tangentium & fecantiurn, Trigonometriz
anfervientes, habemustabulas Aftronomiicas perpe
tüóoruní motuum coleflium, ex qvibus & Ecclipfes
lumiridéium fipputantur & Ephemerides de-feculo
fid fcculum excerpuritür pro Caléridarns ex iisdem
porre
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merifecum & multiplicari & dividi debuerint, fummoperé
de compendiis follicitifuere fummi Viri; Mirabilerh inven .
tionem füb nomi doftrint' Profthapheretice exhibep-
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plitare aggresi futt Crugerus, Vlacqvius &alii cireumfej
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fabulz qvadratotnm haud minimum preftare ufurn à. mul,
tisabhinc annis ab(ervavi, easdem veró à Magino & Clavio
89 rediceóp 19990. asqve elaboratas & ad radicem irioo. Vs
qve ab authore incognito manuals Matbeitadici argentos
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'8. 4. Qvomodo itaquefingüke radices db 1; usque: ad Redicet
160600. in tabulis quzrantür, per fe patet; primó enim ad 2vemode
millenarios deinde ad centenarios in fronte, ad denarios 7277/75,
veró vel fimplices, nimirum qui funt infra centum ad latus ^ P.
finiftrum refpiciendum eft. E. g. querenda eft Radix3ó605ih ^^ -
fronte invenies primó 3600 & in latere invenies;, hocfere. :. ^^ -
modó : [i6oo: Iteni quzrenda eft Radix 365253: evolvendo — —
ss t Aad |
& revolveido, tandem inveniesin fronte 36500 & in lare — |
a3. intellige : |óreo: Sic quzrenda, in:tabulis eft radix -- -
t 137 — |
30300, in fronts videbis fuo locá 20 joo & in latere q, ita nij» )
mirum: ! [re j00 & fic de cateris, |
o
$. 5. Ad faciliorem quadratarumEvolutionem (ex pq» Cue i» món
flremz numerorum quadratorum figurz femper funt ex- /*qvadne-
preftz, antecedehtes verà cütn pluribus numeris quadratis f^^» ne
communcs fint, non nifi femel expre(T» funt, ita tamen;ut ing fem i
proxime fequentibusetiamtanquam expreffz fubintelligan. omis fr.
tur, usque düm adalios prepofitos numeros expreffos per- ri eonrinud
veniatur. | | ferte expri-
6.6. Hinc quilibet quadratus numerus facilà evolvi. "tr ee 0.
tur, abícindendo nimirum per interpofitum punttum vel Quadrari
comma, 6 pofteriores ad dextram numerorum figuras ; fi pie i
hoc fieri poffit, hac caf enim aut nulla numerorum figue T uA 4.
ra, aut unica, aut duz aut tres, aut quatuor füperfünt, qui-zemsr?,—
bus fingulis cafibus primó feré tabularum intuitu invenies, (Primé)
ubi (imiliter vel nulla vel unica, vel dux vel tres vel quatuor ab[cindendq
numerorum figurz predictis 6 ultimis fint antepofitz ;Non [ex a as
" minus etiam in altero cafu, idem evenit, quó fex figure ab« e RAUS
(cindi non poffunt, quando nimirum propofitus vel datus
numerus quadratus pauciores fex figuris habet ; facillime
| B enim
e
LO Q0 EAPLH.
.. ,- enim téperitur tabularum . locus , ubi vel ex unica vel ex
, duabiisvel ex tribus;vel quatuor vel quinque numerorum,
: figuris quadrati con(tent numeri. —— |
(Sicundi) &. 7. Factà tali abíciffione initiales ad (iniftratn nume-
imiti«les — Ti omnium primó confiderandi, & in tabulis querendi funt. —
venfidenendo [ucceítivé de proximo ad proximum pergendo, usque dàüm
€» rabnl namerusin tabulis cum dato feu propofito aut exáàcté cor»
^ ^ reípondeatin humeris exacte quadratis, aut (altem omnium
aninorum maximus, i,c. Dato numero proximé mimor in
-numeris exadhà radice carentibus, reperiatur, E. g. queren-
Pepe dus in tabulis eft numerus quadratus 3479112256 ; abfciffis 6
Wltimis, remanent priores 4 nerhpé 3479, hunc nurnierunt,
quzro in iíto tabularüm loco, ubi continuó 4 fünt antepo-
tz figura: quidem ubi umerus initialis eft 3j. poftmodum,
. facillime invenitur, ubi [equens fit 4 & porró ubi fit 2 & nbi
9. Hifíte 4 inventis reftat, ut abíciffz figure conferantur
cum ó figuris in tabulis five cum antecedentibus 4, conjun-
€tis fivé proxime fubjectis , in quibus omnes 6 nempe r2256
exacte in tabulis reperiuntur, & quidem conjunctim cum;
. fuis 4 antecedentibus, EM
Exemplum Itém quaratur e.g. $j1, $4189 1. abfciffis 6; remanent 853
Sechitidum. quem numerum in tabufia invetjes, habentem 6 adjunctas
. * . figutastempé 6534025. & de(cendendo., ftatim perveniesad —
abíciffttm exacte, nitnirüm $4188: I
Exemplum Potró qugratut quadratus nurnerus 45500496. abícif-
Tertium, lis 6. pluribus comtiiunis4U, 1teer alios etiam exacté fub fe,
| habet,5050496, quod non potet non (ttim inveniri,ctiín hic
etiam initiálesjó celeri curfu ad fuum competentem locum -
- deducant qu&rentem. ME | |
Esmbhno —— — Tandehie.gt. qu&tatur nürértis quadratus $1076, fine
QQuartur difficultate eftendetit tabula, ubinatu lecorum $ tannim.
! fint
4
€Ab 1h
lint figure & ubiearundem initiales fint fubit totus qus
tus fimiliter innote(cit. )
Unicum falrem exemplum proponi numerorum radi- Ex, Join:
cem quadratam exadié non habentium neceffitas ordinis ^s zn su.
mero exalid |
, exigere-viderur : Quaratur- numerus 7069500176 , abfeifis 97^ eÉ-
6 ultimis in tabulis dantur antecedentes 7369. communes
«um adjecto eidem & quinque aliis fubjedtis numeris qua-
dratis inter quos unicus 364015 dato soor76 proxime minor
e(t & dati loco affumendus, in finem detertninandz fractio- -
nis Radici quadrata adharentis ; ut infr explicabitur. — ^ 15, eu, a
6.8. Difpofitisfic & Radicibus& quadratis Gnguls ra- dicus cy
dices refpiciunt ad fua quadrata & vice verfa , quadrata ad 4v4draro-
fuas radices; Nempe : rum insrbg-
$.9. Sidetur aliqua radix, iftius quadratum deprehen- "* "9few.
rente,
ditur in communi concurfu, quando in.tabulis à millepariis Radici dà-
& centenariis Radicis datz numeris defcenditur usque, '* yin
düm aliquis numerus quadratus fit é regione numerorum fuomodo
Radicis infrà centum in latere» Vel brevius: A fronte in ea- quentur ?
dem columna defcende tàm diu usque dum pervenias ad^
numerum qui é regione polituseft illi in latere, E. g. Ra- Es Primsm
dicis 236 quadratum eft 55696. quia in tabulis, tam diu à cen-
teriariis defcendimus usque diim 55696 fucrit & régione i6.
fequenti fere gaudet figura. ||1oo LM '
36 1155696 (0f Ln
Similiter'quadratum radicis too nemp? : 1446000: có- Ex. Stdtnb.
dem modó queritur; in tabulis cnim videbis : |l 11Go dum, .
144.0000 (o '
QNadratum radicis 15962 fimiliter invenierur & erit ^ T: ere.
674, c15444,nimirum in cabulis videbis qu Ui | "oc
674015444 7
$. 10. Vice versà dati alicujus & in cabulisíecundum $. D«té m
| Bz [extum
4
l
!
IT | * 4. 21. |
Alntó, q90» Sextum & leptimnta hujus. cap, evóluti "quadrati invenit
medo Pres, tut limiliter radix, afcendemdo nimirum usque ad Radicit
Tar EM 4o» millenarios & centenarios in fronte & verfus finiftrum,
Aux? —latuspergende usque ad Radicis numeros , ? regione in
* ' latere lümma enim horum omnium erit quae(ita Radix.
Vn Primim. Exempla lint eadem cum precedentibus, quadtati 5,655696
Radix quadrata erit 25$. quia afcendendo ad frontem usque
«*leprehenditnuszoo & & regione ver(us finiftrum latusadex-
Yremitatern usque ypergendo inveriimus35, horum fumma;
,, $36 eft quod quaritur, quod ipfurn etiam patetex appolita..
xlelineatione, ] | : 160 p
| 416 556551 m
- . ^" ^. L . M -w.
T Radix quadrata erit 1200. quia if»
aExSNea Qvaarati1446050 q erit .
ew» — lumüitateeft poo &ad [atus o. erpó eciam fumtra1z00, elt -
7 dpfaRadix,utfimiliterex figura patet. IB IECCNNE t
Wsdeue — Ytaetriam quadrati ó74,025444 Radix quadrata eftz5962.
. — swfcehdeenim & invenies 25900. ac latus refpice & videbis
, Uo utoftenditfigura: ldmee—4 |
(oos pahe Dale 09491624 0154441
"wr —— 5o Utautem innotefcat, quantum compendii affe-.
*ompenilium rdht Xabule quadratorum in fimplici quadraturz negotio,
vf wiliie (dealiisenims& compendiis& ufibus irfrà dicetur) opere
Mortis "reri melt vulgarem modum arithmeticum in iisdem ex-
qu... *mplismodópropofitis fübjungere. |
qi)pergoa- |. Daturex, gr. Radix 256 & queritur per modum Arith-
vs" meticum vulgaremejusdem Radicis quadratum; quod 3.
| DE aré 3- p(amexpediri debet multipliatione.radicis cum femet à-.
- X " * .
t
4
Ergo
t A P. m
Ergo operatio erit talis: miltiplicationis preda £x deii
r
a5
T fextuplum a1416 d&umefts5ó96.idem -
uultiplicandi$ triplum 708 cumquadrato inta-
duplum 471. bulis reperto.
Summa .. $5696.
Similiter datur Radix 1200, cujus quadratum per £x Sri,
vul garem arithmedcam quaritur,fequeri moda.
( Biceutuplum 0. 240002
i----- (implum. . (0
1440600. .'
productum idem cum quadrato in tabulis inverite.
"Tandem datur Radix 75962, cujus quadratum per vul--
|» Saaremaridhimeticam invenitur hócmodó látis quidem -
«liosó: ,
mulüplicandi
4. Tertid.
2y96523
| 0007 25962
uu Tduplum ,....... 5 : $1924
. Afextuplum ... .. 155772
multiplicandij moncuplum « . . . 233653.
| |] quintuplum.. . 129810
(duplum... .. . 519124
Summa ... 71674013444
quadratum quz(itum&idem cum illo in tabulis
. multó commodius
$.m. Ad extrattionem radicis fecundum vulgarem, (2) Prrex-
aritimericam ratio ordinis nos perducit; cujus Regula:cR : edicion
- EI cH amd
vulqavene. figura
e 20 C AP. 11,
fit intermedia absque puncto,stque tutic initium ope-
Mritbiicti^ vationiserit ad (iniftram , ubi prima numeri figura puncto
$4N.S c
notata, unà cum fua antecedente,non notata,(i quam habet,
quariturinfübjedta tabula A. autin fchedula aut in men-
te affervata, fi veró in tabula non inveniatur, affumitur in.,
eadem ille numerus, qui proxime minor eft; & ceu quotiens
in divifione poft lunulam ponenda eft ; hujns in quotiente;
pofita radicis duplum, vices gerit diviforis fequenti figura
.punctó nom notatz fubjiciendi, ut emergat fecundus quoti-
ens, poftlunulam ponendus, facta itidem fübtractione, ut in
' divifione fieri fólet, ejusdem fecundi quotientisquadratum.,
ex fuübjecta tabula depromendum, à fecundo puncto (übtra-
hendum eft ; Porró duorum numerorum in quotiente con-
junctim accepterum duplum eft divifor fecundus fubjiciene
. $usimmediaté poft punctum fecundum, ut emergat quoti-
ens tertius, cujus tertii quotientis quadratum fimiliter fuüb-
trahendum eft à puneto tertio, & fic porrà fi 4, y. 6. vel plu-
ta ctiam adfint punta, usque diim ad finem perveniatur,
Radix, quadratum.
Yo»... 1- ;
2 ..5...4 |
03 9
^. 16.
. . . 2j
6 . . : 36
74... 49
8 .5.4..64
8: $t oci: 8i
veniam Typographo Cyphre diffe & in vulgari di-
vilione ufitatz non fuerunt ad manus, modum dividendi
" minus
em
CAP. IN !
minut ufitarum, commodiorein tamen & meliorem adhibe.
bimus. Datur itaque quadratum 55696 & quaritur ejus- Er, Primé,
dem Radix, fequenti mE i
2 * (n6 Rodi.
quadratum .- 1
4 Divifor Primus
divif tripl, - " ]
quadratum. -. 4 -- "36-7 Divifor. Secundus,
*divif fextupl. .
3
quadratum --36
Similiter datur. quadratum : M4btoo & queritur Es frrindi
mms
4 W oo0o 200 * - Radix,
quadratum - - c
^e Divifor Primas,
divif, dupl. - - -
"$$
auadratum -. REI ;
Nota: qitoniam bicin feeunda dant :
operatione áppartt, nil nifl duas 9 iti;
g& uotiente fubíécututás effe, fécundus
tertius divifor fraffrà quafitur.
7 Tándemi
d CAP. If.
ExTotié. . :.— "Tagdem datur quadratum 674025444. && quabritum
EE ejusdem radix quadrata. EE
. 6 als ala 4 (e - » Radix,
quadratum - -- 4 . ioc
[4 Divifor Primus,
divif. quintupl, ---21 i$
m . 2 .
quadraum - - - | 5o Divifor Secundu:
: a
. 499 t VL
divif. soncuplum -- - 45 dEES ove
pr 2 18. Divifor Tertius,
quadratum - - - - 81 59 ]
215| —á—ái
diviforis fextuplum - - - H o8 | 5192. Divifor Qvattus.
1074
«quadratum - - » - - 3á .
103 84
diviforisduplun - — - 19] i4 .
quadratum - . . 4
c .
Tabularum $13. Satis fuperque.ex precedentibus dijudicari pof- .
* eompendiwm (e exiftimamus, quantum & tadiofis operationibus & cem-
Bn confr- poris difpendio hz tabulz conferant remedium, przprimis,
Qccafionalis càm ej*modi operationes Geometris fint quoridianz, quàm-
memiiode Viscnim alio etiam modó ex noftra inventione & multipli-
move dw- ' Catio & divifio & radicum extractio ad fimplicem additio-
fboráin- nem, fübtractionem. & dimidiationem utiliffimé ita (int
bili 4- redacte, ut planénon opus (it tabulà Pythagoricà, hoc e(t, ut
jon dio; neceffe non fit vel memoria: mandáre vel etiam uti ifta ta- .
fenis &f Ra. bulà in qua habetur, feme] unum;ba8 Cintmabl etrié/ucali-
dicum ext - - quo
ionis,
Y
ey | : |
| |eC4R Hh MM -
quó modo colligi potéft infrà ex $. r1. capitisIV. nihil omi-
nis tamen , tabulz quadratorum compendiorüm ftudiofis «^ ^ ^ '-
lisdemque deditisnon' minus etiam arridere poffunt, pra
primis cüm accedant alii ufüs & peculjari ingeniofiffimà
fundamento, füper quo conftructe funt, nitantur, de quo
fequenti agemus capite. ^ |
P
CCAB HE
De Tabularum fundamento & conftructione:
|. $41. Fundamentum conítructionis eftprogreffionis A- Fundamen-
rthmeticzbeneficium ; . — fum qvod s
^ $. a. Eftveró progreffio arithmeti T js»
$. ró.progreffio arithmetica continua feries Pregresfo
numerorum, zqualiter fefe excedentium, vel, duod idem, 4,/ppurtica c
equaliter differentium ; Sic ex gr. in numeris 3. 5. 7. 9. ri, eft qvi2? :
progreffio arithmetica, quia major proximé minorem can- m
tinuó excedit binarió vel , quia differentia femper eft equa- T
"lis & hóc locó a. " 03 8 NU —
$. 3. Verüm enim veró, univat(aliter duorum quadra-
torum differentia (emper eft duplum producti ex multipli- Du
catione Radicis minoris cum differentia qua eft inter radi- 2n diffe
- ces, majorem & minorem additó infüper quadrató cjusdet resria av«-
radicum differentiz, per prop. 4. II. Eucl. atque fic, fi radi- sm ?
cum differentia eft Unitas, (1) quadrát. majoris radicis differt
five ditat à quadrato minoris radicis fecundum differen-
tiam quz fit ex additione dupli minoris radicis & unitatis.
Utrumique exemplà illuftrari neceffe eft ; Sint pro ju-
bitu duz radices 317 &423, quarum , differentia eft ur, füb.
. 428
trahendpnempe minorem de majori; 317 Qvadratum ra-
cC dicis
£x. Pri-
Stute.
C4». INS
dicis | 7. eft 160439. *& tjuadratum rai icis 413. elt nya,
(quod utrumque ex tabulis liquet ) quorum differentia eft
£695, itidem perfubtrattionemuminoris quadrati dema-
jres 2 1831.83.13. |
190489
Qo. $1855 HN
. Dico; hancdifferentiam:g:695 conftare ex duplo producti
*x multiplicatione radicis minoris 317 cum d
entia radi-
317| *cumnrmetnpe, 70374, addito infuper
—LLU] wuadrarbejedem radicum differeriie
y : 7 "myquod in tabulis erit pr, Summa.
- 317 enim erit differeritia quadratórum pra»
mepastestentutilb e 2 . .
35187] dica, 82695, ut patet. .-
5197 !
$0374 ;
3 2.32311!
2698$
Oe. Set" — "Deinde, fint dpz radices 3t7.:& 313. quatum differentia eft
ven. unitas, fübtrahendo nemp? minorem de majori ; minotis
Es, Torri-
AE
quadratum eft 100439, majoris veró rorma , quorum diffe
rentia per fubtracttonem minoris de toajori eit 65, Dico:
anc differeritixm confillere in duplo mino-
| in ris radicis ój4; (ceu duplo produdto ex mino-
C $9] ceradice &ditferenia, quenonmultiphcat)
635] - -adjectàsunitate (ceu-quadraró differentiz x.)
"ut eveniat predicta sjuadratorum "Summa 65.
utpatét — | | |
317] . Metnerit;in quibusliberexemplis;in quibus radi-
— ces unitate differunt, e.g. evolvanturradices;719.. -
633). & 5715). illius quaditatum e(t 326655104, 'hujus veró-
i|. 3166465409, dico ; differerttiam horum quadrato-
63
Tumi4joy elleduplurm minorisradicis, adjectà .—
| 27 anfuper .
C 4 P. 111. Ip
infuper unitate, — 3366465409] . $7154
3266351104 : Y
114305 114304
X
6. 4. Hóc prafüppofitó latere quidem jam non. poteft,
ratio con(iruQtionis tabularum à Radice r, usque ad Radi-
cem 100000. imó continuationis earundem iri infinitum, (i
placeat ; Sed nondum apparet, quomodo fecundum dicta...
. in$.r hujus cap. fundamentum conftrudtionis fit progre(s
(ionis arithmetica beneficium, quod porrà videndutn. - u
, S. s. Univerfaliter omnis progreffio arithmetica itas
affecta eít, ut ejusdem equé multiplicia fimiliter conftituanc Pregref- —
progreffionem aritheticam , alia tamen differentia pro-9» bos
greffionis gaudente, nimirum illà, quz fit ex multiplicatio- Giendli f
. medifferentie prima progreffionis cum illo numero , per eva,
quem altera progreffio etzqué multiplex,
$. 6. Exempló reserit clarior; ponatur progreffio arith- Ex. Pri-
metica3. 6. 9. 1. 15. &c.. cujus differentia eft 3. dico; hujus men |
progreffionis zque multiplicia rj. 3c. 45.60. 75, facta ex mul«
tiplicatione cum numero ;. fimiliter conftituere progreffio-
nem arithmeticam, cujus differentia e(t 1j. facta nimirum,
ex multiplicatione differenti priotis progreffionis cum nu-
. meros. per quem altera progreffio eft zquemultiplex ; Sic
tadem progreffio ;. 6. 9. 11. 1. multiplicata equaliter & fem» Es cin.
. per cum numero 2, exhibet aliam progreffionem arithme- dum,
ticam 6. 12.18. 14, 30. &c. cujus differentia eft 6, facta nimi. -
rum ex multiplicatione .differentiz prioris progre(fionis
cum numero 2. per quem altera progreffig eft equé multi«
plex& fic de cateris. 0
$. 7. Demonftratio hujus rei eft talis; Ex definitione ,,, enin
. Progreffionis arithmetica in $. 2, tradite patet,candem con- '' (ie.
ftare excontinua equalisexceffusadjectionejponaturitaque ^
primus progreffionis arithmetic numerusceu primus A
ZU 2 207 IRunD
-—
-
E
*
| 20 | C AP. 111, 0.
minus íeorfim ità, üt fecundus terminus conítet ex primo
& Exceffu & tertius términus ex primo & duobus exceff.-
bus/ffimiliter etiam quartus terminus ex primo & tribus ex-
- : 'ceffibus fingulatim ; atque fic porró in infinitum; Jam.
veró omnium iffoórum terminorum fi fumantur equé tmul-
típlices, ità quidem , ut tàn multiplex femper fit primus
terminus in quolibet alio etiatn termino fippülatim pefi-
/ tus, quàm multiplex eft Exceffus five femel five aliquo-
— gies in aliquo lit pofitus termino, erunt etiam perprimuam V,
| €ucl.&qué multiplices termini progreflionis füummatim ac»
| cepti, à quibus fingulis, fi auferas primum terminum equé
' . Amultiplicem, erunt refiduorum Exceffus,exceffibustotorum
— equalesper ax. 14. Primi Eucl, Scd.tali modo, equé multi-
plicium fecundus terminus, e(t foldm unus exceífus equé
multiplex, tertius terminus funt duo Exceffus zqué. multi«
' plices, quartus terminus funt tres Exceffus qué multiplices
& fic porró ; fed unutn, duo, tria &c, fantin progreffione;
atithinetica , per definitionem d. $. 2. quare etiam hi emnes
termini erunt in progreffione arithmetica & quidém fecun-
! dum differentiam, quae fit ex multiplicatione Exce(fue cum,
|. 70 eópér quod altera progreffioeft zqué muktiplex, Q.E.D.
NEN " 7$.8. Quemadmodum vero íià progretfionalibus arith-
rétrésfo- tiieficé numerus auferatur £qualis aliquis , ipfa progtesfio .
mis deb, — A ut thaneat arithmetica p.d. ax. 1g. áta etiam fiulli pro-
effeiiio, NN DEDE $ . . P
UU gtesioni arithmetic obflat fi contínuó fingulis in progtes-
fióne términis equalis vel idem numerus infuper accedat
P. d. ax. i6, Primi £ucl, Vcita:.— | BE -
Süpilicafes |. S. 9- Univerfalitér in omni progresfione àtithmetíca ;.
dio, filingulistefminis accedat áliquisnumerüs perpetuó idem,
! adhüc illa progresfio fit arithmetica, cujas differentia i!li,
qu£ ab initio £qualiseftp. d ax. ^ —— c | |
TEE $.16. Qvoniain itaque in tabulis feoandum ordinem.
- L 24 Radices.
-
-' €APRIIL 005 3:
""Radicesariththetjce/progrediuntur, fecundum differentiata 4
" Unitatis (7) & fecundum $. 5. differentia quadratorum fem-
per eft duplum radicis minórisunitateibfüperadjectà;per- . .
fpicnum eftnon attentá juxta $. 9. unitate, differentiasqua- ^-^ — ^
dratorum fecundum ordinem Radicum in cabulis adícripto- EM
rum, fervare progresfionem atithmeticam, cujus differen-
£iaeflz, juxta S. y. — ^ ^ 5. mE
$. ar. -Hhrjus rei véritas fortasfis facilius colligitur, fi -
«uis confideret fubjectas figuras, quafi ex radice unitatis or»
tas, & fecundum unitatem fuccesfivé quoad radices auctas 3.
, Vinicuta enim primó erit punctum, fi tanquam quadratum
cóncipias unitatem, jam veró fi pergas ad radicemaz.quatuot
habebis in figura quadrata puncta, ergó differentiam ho-.
rum d&orum pritmorum quadratorum conítituent triz.,
puncta ; Pérge ulterius ad radicem 3. & videbis antecedenti
quadrato radicis2. accedere qninque.punéta, differentiam
quadratoruniexradicibusz. & 3. conftituentia ;'Sitniliterad
radicem 4. perge & accedent ceu differentia 7. puncta & eo« tC
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turi d, e. f. g. effeminus quoddam quadratum, & cujus ra*
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titudo fimiliter cadem eft, xempé unitas, quales funt lineae.
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radicis quadratz, fi nempé dato quadrató ejusdem quzratur
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' per additionem quadrüpli & fimpli; Sextuplum per addi-
tionem quadrupli & dupli ; Sepdaplum aut per additionem .
quadrupli,dupli& (itmpli autper fübtractionem (impli ab
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Hic veró notdndum, quód propter defectum Cypbrarum...
diffectarum,nümeri füb quetiente áliquos (apra (criptos ha-
bentes, prodiffectis, ficut in vulgari divifione fieri folet, ha-
beri debeant, & tantüm füpra &ripti valeant, Poftquin-
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& initiales fuiffent 24. qui fubtrahi hon potüiflentà a; ; Se-
primus quotiens eft 5. hocipfüm ftatim fignificant initiales
quadrupli 15; nam fi hos'addas dupli initiali 7. habebis 22,
numerum majorery initialibus in füperiore à quo fubtractie
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fieri per additionem poteft immediate fummam in compe-
tentem locum transferendo :' Octavus quotiens eft 2, (ta-
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fuperioris in fübtractione facienta eítg. Ergó ipfum du.
plum, cujus initialis 7. eft proxime minor, qui fübtrahi po-
teft: Nonus quotiens eft 7, cünij dupli initiales fint 74. fupe-
rioris vero in fübtractione facienda initiales(ünt 74. Ergà
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usque ad finem progrediendum & tandem retrianebit nu-
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logiftica decimali, cüm defint typi idonei, arithmetici libri (5) fecus.
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duó, radix extracta habens fractionem adhzrentem án pat4
tibus Io. Iooo, vel sjoooo. eric. préximé véra, quod exem
plis éft illuftrandutm & quidem prinio, in numcris qui ilu
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iidod Pe binis Cyphrisejusmodi numerus vel faltem proximé minot .
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radix »« - ,214*.— ! |
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eujis radix elt 2.0 9 52. quapropter radix numeri 4589 9-
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àmeroó - - 5338.040000, quod. ibidem:eft - - 5337:909721. cujus
radix eft - - 706. quápropterradim numeri ri59lo4- tic .
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' Cypbrisin tabulis quaritur quadratum proximé minus nu-
mero - - 697.099300., qupd ibidem eft - - 696960060, cujus
radix eft 16400. quapropter radix numert -- 6970603.
, erit -.» 2640. 9 hoc eft.- - 2649. absque fractione, nifi ves,
P (ej
juxta $. fequentem extractionem- continuare. .
Ki,No"u". ^ 'Tandenrextrahatur radix exnumero - - 8572975. Ad-
jedtis duabus Cyphris,in tabulis queritur quadratum proxi-
mé mirus numeró - - $576297500. quod ibidem eft- - -
8£157.122446. cujus radixeft go206, quapropter radix numeri
4537297f.€rit -- 9o20 6 Continuari verá euam poterit ex
io.
tractio fecundum fequentia,
Biecafommul'ia 6.27. Singulari ftudio in precedentibus Exemplis Q-
gabularum , Yes exhibui cafus , qui in fimilibus occurrere poffunt in.
ercommep-. quotidiana praxi geometrica, cüm in ejusmodi cafibus in-
! 45
Es. 084--
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dti. ——— credibilecommodumaccrefcat per haíce tabulaspraxi Geo-
metricz,ut nullus dubitem, ob hunc folum ufum, nullum,
. fore Geometram, qui tabulas tetragonometricas de mani-
bus fuis dimittere cenfuültum judicet , reftat jam, ut per e»
6 ip same. xempla illuftrentur illi cafusin quibus extractionis conti-
pis tran[cen-
aentibns il- Quatie transgreditur numeros in tabulis contentos,
dor 42: funp | Extrahatur radix ex numero 375974210. in tabulis pro-
ént5x 9. xim& minus quadratum e(t - - 375.972:00. cujus radix poo,
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t ultima fübtractio pro refiduo determinando;qua fieriali-
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xis compendió, ut ez fuperiori exemplocolligi poteft.
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temporis tum laboris compendió & quadrars, & er illis ea-
dices extrahi poffint, id quod tabularum primarius eft ufus, .
fequitur jam fecundarius ufus in operationibus tànx arish-
meticis tiir geometricis, ubi primó de applicatione & ufia
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ritkmnttic4, tur, in maximumrei Mathematice detrimentum,cüm tabu.
lam pythagoricam, i, e, Semel unum eft unum &c, memo-
' rüà retinere non poffint nec, etiam aliquando ob imagina-
riam difficultates veliht,quod tamen communt arithmeti.
cz medó necelfarium eft, & infuper quamvismemoriá idem
.. fit retentum, tamen adhuc operatio ipfa nimiamrequirat &
; " gttentionem & patientiam atq; pretereà multuro confumat.
temporis, varia propterea à praftantiffimis Viris excogitata
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lorum nuraeroruni,fi j paresfint,. dimidium folummodó füb-
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quenti can jungatur,(imiliter fic fubfcripto dimidió e. g,
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^
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Differentia inter quadrata dimidize muleipticindi &
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à. "^ $h
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| differentia eft productum - « - 43637127.
Ydem éxemplum vulgari modó elaboratum eít in Ex,
. 1L. 5.3. capitis IV.
Xem ba, — Sit nümerus multiplicandus 1aégr. Malriplicans 3769.
t Operatio erit talis :
-mult&plicandus -« i636; "
muliplians « « 35769 -"
Summa: -- 199406
dimidiam - -' 9970 jhujasquadr.$946. tiios !
fubrradtó maltiplicante - 639 34 huj jus quadr. 4087. 4087.53 6556
differentia e(t productum «2443313185
pia Ca $. à. .Si veró füumme ultimué ad dextram numeras fit
4$ d pecedi ;4/, TÜPatS quedidem raríus priore cafü accidit,tunc ablatàá
| ame »341. fumma ünitáte & éruce memoriz ergà appofità , operatio
- tiplicamhi 8 cum pr&cédenti quóad cetera plané eadet eft fed à àn fine,
jsntiliipli- ^ "taultiplicans nümerus adhuc addendus eft,
, N4htistl-— Sit nymerus multiplicandus - - 8 ii, & multiplicans 19234.
inu Ad "Operatio erit calis i
estem eil quktplicandus - - $5219
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forse mulüplicanse 32742 - hujusquadtat. 1 071.038 $64
difzrentig . « 1681652014
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) producta duarum, trium vel plurium multiplitatipnum.,;
quot enim feparationes, tot erunt multiplicatipnes, 0
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cans -- 27891. Hoc exemplum preut feparatio mulüpli- 1 re ribus
candi numerorum libuerit, tribus diverfis medis elaborari Ziyerfis won.
poterit, aut enim ad dektram tres nümeros » « 479. ast die elabenoo
quatuor 3479. aut quinque 1479. feparare poteris , femper /9e,
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quinque & propter extremum quinque Cyphre (o)
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operatio quintá erit « « fqpip2400so00002908.
Extremus ad finiftram oem rà,
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que;Cyphre (0)-&ita precedenti producto 5352150.
accedent quindecim Cyphra (o) & erit fic: ——
Dperatio fexta» - - £52j218 £0000000600000000
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Dper atio tertía LRL | 43640:1990000050000
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pe . li cafu valgerem multiplicatianismodum & non prolixum
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Ma a nolim :demonftraffe;pro cvitandis erroribus d; 10llenda neceífitace
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&. e. per femel àónum e(tunum, &c. quadratorum canonem
titilem elle.
S. 8. Reítare videtur demonftratio mulüplicationis im-
DMsltiplica- jus noítrz, ad fatisfaciendum illi, qui forfan earidem défidc-
mea rabit, defümimus autem illam ex propof. ;. r1. Eucl. ubi de-
monitrat : 5j Feifa linea fecetwr in equalia € non aqualia v,
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Week IM, vi, quod À dimidia deferibizur. quadraro, —
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"e — quales partes a.c. -& c,b. nec
r noh induas'inzquales par-
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pter intetmedia harum:.dua-
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"dangulum fub fegmentis inzqualibusa, k. i. f, unà cum qua-
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"deferibitür quadrato, nempe b.c.d.e.. |
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uppllvittio, "ütértudjorenr multiplicandum & lineam k.b. eífe aliquem,
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intermediam c. k, effe differentiam, (i à digsidiafummaimis ^ ^!
norcm multiplicantem auferas vel fubtrabas fed; qpadsas —^ uu
tym dimidiz fumma cft. zquale, redtaggulo; fub ipgqualis —,
bus.& quadrato intermedia.ceu füis partibus fimül fümtis,, "E
ablatáitaqueuna partequadrato nempe intepmediereftabit — -
altera pars nempé rectangulum (ub inzqualibus.i, e, multis;
plicationis productum 5 Q. E. D. B eod
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jus.dimidium b.-Fc, áquodimidiofifubtrahasmulriplican-. — — -
2 AN "M
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. 4
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rei reddimus rationem , quód in quovis multiplicarjonis r4» f/ sfts-
producto totiesinfit multiplicans minor quot in multipli- fien merMC
cando majori numero funt unitates , quare fi. unitatem &iypup,—
multiplicante demanus faltem in mente. & fic qpultiplica- mE
tionem perágamusjid efficiemus, non folüm;ut fumma mnl- vod
tiplicandi & multiplicantis non amplnis fit numerus impar
fed etiam, ut productum deficiatipío multiplicante nume-
ro , addatur ergà defcctus & erit productum completum.
Q.E.D. Hz ^, . $1j.Utrum
! d CAP. Y; | S
Jy Iioilone 8, Wy. Uttiti etiam circa divifionemtabulz tetragóno-
Yr vn metrice aliquid afferant. ntilitatis vel compendii jam vi-
| per dendum, &fciendum quidem,ficut in vulgari divifione quo-
axbela dw. ticnshaud fincanria inquiitione an procedat, nec ne? fin»
wewiPi paf- gulatim invenitur, fic etiam quidem per tabulas hutneri in
Jm. 'quotiente non fittül omnes fed bini & bini inveniuntur, fed
ultró apparent fimplici additionis & fubtractions benefi
ció, fingulari &atiliffimo & faaviffimo compendio , ut fe-
«uitur. — | Dot MEE
pubp $.14. Indivifionenoftra per tabulas tetragonicás duo
aibilar, futit diftinguendi Cafus; Aut enim divifor minor eft quàm.
quindo 4j- 99906. aut rnajor illó-numeró 99900. Sidivifor münor fit.
"wifor eft mi- rurfus fübdiftingur debet; aut errim divifor habet. quinque
40r 44" numéros aut hacet pauciores, — mE
vo. $15. Hoc cafu divifori adjicienda für cot'Cyplirz nul-
58i páncioes litatis, quot divifor quinque numeris habet pauciores ; ex.
«livifor b4- gt. (i divifor fit 559. adjiciendz (uttc dus nullitatis Cyphra,
vot, uim. , Dt fiant quinque. numeri; hóc modó: 35900. fimiliter, fidi-
'gain2ae, de. Vot. fit 9. adjieiendz lur quátuor Cyphre hóc^ modó,
et i. die. 90006. & fic, fi divifor (it 25506. erit adjicienda una: Cyphra,
iu Qypbrie ut fiat dwi(or --2560.&c. "Quotquot veróà in quolibet E-
gutes n" xemiplo divifori adjecta fuerint Cyplrz, tot etiam.unt ad
| quA "i - Jiciende CypHre dividendo; Quo factó hic cafus adequa-
tuseritilli, quó inivifóre'fürit quinque numeri- :
aS wiitiferis rua i db E
Sie :$. 16, Adzquacds hifce duóbus cafibus & inunum reda-
fntmajores CiStafüm, rutfüs átit diviforis initiales ad finiiram funt mi
quáminitia- nórcs aut tiajores imitialibus numeris dividendi; Hóc cafu
e t, r« n dividendo numero à finiftra verfus dextram feptem funt,
zprem; 2$ 24, Utr iiterjectamitiàeblam abfcindendi numeri, illó veró cafü
ed. frr su Cent Ex, ficut etiamrin viülgari arithmetica divifor, cu-
Afunt ab/ciw- !)sinitidles func minores ihitialibus dividendi, totà divi-
«eie j 71 NHNC« . Dot . UO M - ^l .-- Tx. . P de
"Pb ee ido
at —
-—
BEEN J7)92 200 *Àygee? fum
: Wlendo uhius quotientis gratià abforbet Tiufmeros , quot i- miverer
ple habet, & in fuper adhuc unum, fi initiales ejus fint màjo- Aexriow fex
wésinitidlibusin dividendo. In divifione'autemmnoftra ad fmt 4H e
liuc unus mimerus accedere debet, quia duo quotientis nu- 4, focur c
4neri fimül prodeurte, |. 1 — Sevmlgari
* — $. 17. Tali divifionis przparatiotie peractà, dividendo mifesc.
fübjiciatur ejusdem duplutn, ab(ciffis fimiliter 4b hoc:duplo fp/a divifie--
illis numeris, qui aábíciffis fünpli fuppofiti fant'& porrówis epemie
fubjice hifce ab(ciffis quadratum diviforis, cujus quadrati£/? A
fex ultimi &iumeri ad dextram , db ántecedentibus ad fini pu euh 3
tram, Conmrmate vel puncto firit'interdiftindti , jaxta $.6« M uad a
*Cap.2. tque jam examina fümmain diviforis & dividendi (rosa.
. 3lupli, aft non integram, fed folüm illam, qux fucura eft, in, 444endo;
ihitialibus ante punétum vél comma numeris , &igfupér iti) fuemam
«duobus poft punctum vel'comma fequentibus, quatn fati borum exa-
atiam ad inrerimrelictó[patió irttermedió fübfcrrbe, & eni-minemo,.
nus follicitus de quatuor numeris fequentibus, qualesilh', , ,
HJ lÉ Misc -—— LA T2 ] (42fummo,
pr&cife futuri fint; iisdem quare in rábulis quadratum pr quartum
ximé minus, in numeris faltem initialibus, i(tis'fcHicet, qui 2754/52 »i-
funt ante punctum vel comma, & duobus'feqüertibus atque» quaeret
liic duofünt diftinguendi cafus; Aut enitn duo poft punctum doi bulis,
vel comma feqüentes numeri, in fumma ad interim addita(j) ete»- -
differunt foliim unitate à ducbus póft punctum vel comma derdend m
mend NP RT, s.a Ql A ferentia.
riumeris in quadrato dictum proxime minus m cabulis WI, spitire
/"mediaté Tequenti aut differurt ab iisdem majori quovistus,,;,. uu.
mero ; Hóc cáfu quadrati proxime minorkf'in tabulisesferiepe 4 074
Sbantur feorítur radicis.duo ultinri ad dextram ; qui femper (47 májer
in.tabulisfüntia marpime, & ab hifce duobus ultimis exfcri- 45 leis —
otis, Tabtrahantur duo tiltitiii diviforis nürneri, five His'ex- ác fib |
Jeipeifinc minores five májorts;freaim fineminores, eó dn ds
xandum ét, quod ab antecedentibus radicisdenariusmutuó diviforit,,
MN 2 (42 m fumi ow
!
e -
| 44$ CAP.PF. — |
fumi poffit, ficut in vulgari fubtractione denarius mu
"^ abantecedente numero fumi folet , refiduum liujus fub-.
|o. —traddionis habebit duos primos quotientis numeros ; llló.
(9) f mint» ver cafü rariori quàmyis, quó duo numeri poft punctü vel ;
ets. has coiia taptüm differunt unitate, vel min" exfcribanrur adhuc:
quadretum. quidem proxime minoris quadrati duo ultimi ad dextram;
refdui addi- fitque fubtractio eltimorum diviforisut jamjam dictum,
tum ef|ci4t (edcentanduns eft, utrum hujus refidui unitate aucti qua-
famimam [ dratum, quadrato diviforis & duplo dividendi additum, ef-
rm ficiat íümmam vel equaJem, vel etíiam majorem, quadrató
dictum proximé minus immediate fequenti, nec ne ?fi non.
faciat z qualem, vel majorem; illud refiduum adhuc habe-
, Mt duos primosquotientis numeros;fi veró faciat zqualem
' vel majorem, quod quidem fítatim apparet per initiales.
. quadrati addendi, tunc refiduunr unitate auctum habebit
. duos primos quotientis numeros, quapropter confultum.. :
hic erit,antequamcertiores hujus eventus fimus,in peculia-
rifchedula hec ipfa tentare, ne feries ipfa operationis dif-
, turbetur; Poftquam veró certiores de duobus primis quo-.
(3) Drvents: gentis numeris redditi fumus, illorum quadratum diviforis,
quo reri quadrato & duplo dividendi in relicto fpario intermedio .
$'ud. fübjici& cumillisaddi debet, fümm&horumtriumrité fub-
| .. fcriptà vel potiüs refpectu numerorum jam anteà fübfcri-
(9)quadra-. ptorum füppletà,À qua fumma demü quadratum illüd quod;
tumproximé ipli proximé min" vel zquale eft in cabulis, (ubtrahendü eft;
' minu fuüb- — (ibfcriptó refiduó, cui refiduo pro inveniendis fequentibus
erabend?. — c n otientis duobus numeris,adjiciantur duo fequentes, poft.
| ( te) [quen abíciffosin duplo dividendi numeri vel faltem unus, fi duo.
e vehe ne ys 9i : ; , 4. non füperfint, iftó enim cafu duo quotientis numeri prodi-
jiciendo. bunt, hóc veró unicus tantummodó, atque huié refiduo
cumadjectis fübjice,; ut anteà, quadratum diviforis & exami-
na
l
CAP. Y. *
— Óna,ut amteà, horum duorbm fummam in anitialibus & ad (i) epem
anterim relictó fpatió intermedló fübfcribe & quxre qua- auda Cf "
ratum proxime. minus & plané ut anteà operando, petgé, c u.s
tum proxime minus & p perando, perge, cuem divi-
"usque dàm nulli:in duplo dividendi numeri fuperint, & 9- £u coni
'mnes quotientis defiderati numeri prodicrinc; Tandem, ii »aando.
addicó fecundum:przdictum modum quotientis ulti vil
*etiam duorum ultimórum quadrató, fi quidem: fümima fic
*wqualis quadrato in tabiilis, ita;ut hocá fumma fübtractnin nando
"nihil relinquat, fcito, quotienti nullam adharere 'fra£tie- »u/a 4uo-
. "nemy'fi veró poft fübssadtionem.difti .quadrati'im tábulisArienti adbe- —
fumma przdictanamerus quispiam remaneateritdimidium re4tf7 46:0
dajusnumeri qui-remanet , namerator in fractione quo- Quando ef
"tienti adjicienda, cujus frattionis denominator eft ipfe divi- 124 fra-
Tor & quidem unà .cum Cy phis, (i quas juxtá S. 15. adjettas 7, 2
abet 0 7 77 BEN M
S. 8. Cüm defcriptio divilionis noftre in S. 17. fatta vix Defzr:pise-
"contrahi potuerit, ad.declinanda omnia dubia, quz operan- "4 "fire
ti in re nova nec aliàsà quopiam hactenus pro mea quidem ifevitas
MAMnTOEP d 4 075. Qr teu oro tr c WircA divifio.
notirta, Inventa, per. marginalia-nonfolüm, ibidem brevius sess, .con/a-
"operationenttradidimus, led eciam .demoenftratiopem.illi-4 pere
us divifionis hic fubjiciendam effe arbitrati (umus, cm hac (1) ser mav-
: ipfa démonftratio ad meliorem intellectum , facere yidea- 2/s4/ia de-
, tur Illa veró demonftratiofaum haber/fandapentrm ig. 72907.
jpropot. 4. n. Eucl; ubiproponit& xlemonftrar: Xj zeffa [is (2) pe£4-
0$ s. f Bea a,b. [ecl a At utcuque in b.quadr atum , 7? to-
Nl Oo 04 H quod tora 4,6. ofertar. ne c. d. P aoa
Nu 9. aequale G5 illia, qua d fegenantis.A. b, && bob. ese upe-
| deféridimtur quadeusi d.e. f 3 E b maio
3-448 Mi quad ^n fub fequreatis a.b, 07 bb, nis fuma
d cemprebeuditug reiiapulo. nepepe. a.b frg, mention.
| K-EUTP |
"7 € 4P. F.
6 Ki.c.e. Sit itaque divifor altemutrum fegmentoruot,
quodlibet a. h; vel h.b. & quotiens efto alterum fegmentumy.
Fandemtnti Qumma veró diviforis & quotientis, nempé a. b. efto illa ip(a
pplisdtity dix in cabulis, cujus duo ultimi ad dextram exfcribendi
| . funt juxta regulam, nam integram exfcribi & totum diviíà-
rem fübtrahi proptereà in regula tantummodó non opus
fuit; quia radix ifta & divifor tantüm in ultimis duobus fem-
per differunt, cüm per tabulasplures duobus fimdl numeris
jn quotiente reperire non liceat, fed fi à fümma illa,ceu tota
linea a. b. fabtrahatur una ejusdem pars nempé divifor, re-
linquetur neceffarià altera pars,nempé quotiens, fed & qua-
dratum iftius (umma feu line conítat ex quadrato divifo-
ris, quadrato quotientis & duplo Rectanguli fub fegmentis.
- &c. duplo dividendi, quare additis his tribus, fümma aut ft
equalisaut major quadrato ex fumma diviforis& quotica-
tis, (i equalis, factà fübtractione. nihil reffabit & proptereà
nulla adherebit fractio in fine operationis, fi veró major fit
fact fubtractione & adjectis dupli dividendi fequentibus ex
eàdem[ratione operatio continuatur, ad finem veró perdu-
£tà operatione; re(idui dimidium erit illud ipfum, quod in
dividendo divifore minuseft;ut is adhuc femel in dividendo
contineri non poffit, quare per fractione determinandum
| .. quotientis reliquum ; omnia quidem fecundum tenorem
Conjulipo- Regula. Q.E. D | | )| |
pe Aeui- — 6.19. Éxhacipfademonftiatione fluit divi(iónis noftra .
(), e» Re fequens regula breviffima : Dividendi duplo.in tot humeris
ii ex de- quot ad eruendos unum vel duos fimill quotientes fufficiunt
menfmrie- addatur quadratum diviforis & infuper quadratum futuri
sedem» — quotientis in uno vel duobus numeris, Summa, quadrato
pe — diviforis & quotientis fimdil, füumtormm aut. zqualiserit aut
pujer, ubi adjectis porró de duplo dividendiunó vel dao-
id a MEULESS | ATHEN bus,
| CXEPF,. | dp
butsnumeris códem, ut anteà modó divifio ad inem usquc
'entinuanda c(t. MEE v
$. 10. Quemadmodum univerfiliter qualibet regula. ,,, o. o.
-^ exempli fit clarior, ita etiam multa hic fübliciemus & Emil Men
ctiam unumquodque illorum compérenti (iia explicatione sjemee. «tque
illuftrabimus, ita quidem, ut ex ipfis ftatim exemplis ratioe7fPirsimto —
operationis defümi& intelligi peffit, preprimis ab illis, qui
in ftudio arithmetico & Geometrico peregrini nonfunt.
€. 21. In ipfis veró Exemplis rationem ordinis haberi, f J Exe; )
multum facitad rem ipfam perfpiciendam,quapropter pro- pÍorumor-
ponemus primó Exerpla fine fractione , eademque rurfus Zue ef jud
fecundum ordinem cafuum, tempe ab initio Exempla abs. met bad. |
que fractione,quz habent diviforem, pauciores quàm quin-
ue numeros habentem, quz quidem,(i duos & plures pau. CétPené-
ciores habeant, pr aliis hoc habent prerogative, quód nul- am im i "
la debeat fieri. fubtradio ultimorum duorum diviforis » drvifor Pun.
(quippe funt tantüim due Cyphrz nullicatis) à duobus ulti- oy c5 uni
mis radicis; deinde Exempla absque fractione,quorum divi- * ve! 4woz.—
for habet qninque numeros, utraque veró ita comparata , veltri.
funt, ut quotientis duo numeri absque ambiguitate (tatin, "i "mit vo
appareant, ctm duo numeri poft punctum in fumma injtia- . '
Ji dupli dividendi & quadrati diviforis, à duobus numeris:
poít punctum in quadrato tabularum proximé majori, plus
unitate differunt, pofteà (equentur Exempla itidem absque .
&ractione ubi hzc differentia unitatem nen excedit & de.
hinc abibiguitatem quotienti defiderato parit, tandem E
xemplum cum fractione proponemus & ad fragtipnem i,
numerisrotundis defcendemus,
-
*
|.
, Divide,
4 | DMEWROS 000
Exemplumo — .—— VAvidendits fit y 7 46952. divilor 5. cum a.Cy-
(0) 491gue Teac olco) Pliris 8 o 6 09.
fnittiom in duplum cum 4. Cyphr 1s 15.4 939 L^ 4
2 & diui- qeadretum divi» G400/000:0 004. 1. 1.
; A" divi- - z i. 4.B OM . 20.
hr hi " quadratum quotientis : 9i 16 quotientis pri»
funt. «djici- Summa - o 641.5031 2-0 mus & fecuns»
ende Cy. quadratum. minus ——3:369216| 1| laus 96.
4^ ——— c Reduum cum fequentibus 3 3 9 0400 :
| | raum divi T 6 400.000000| quotientistere .
«quadratum quotientis -c $8 39! leus& quartus
Summa - - - 541x397323| 1. .- - $
quadratum minus -- —— 2363839. |.
rcíiduum cum fequentibus - 11 0 4 oocoquotientisquin-
quadratum diviforis - - 64 06.00 co60|tus & fextus 69.
, quadratum quotjentis - | - — 4761 integer quoti-
Sutra zqualis quadrato - 64:1. 6 4 47 61 ens 968769.
xempl. 2. .. Dividendus fit - - 3507.8 »ola divifor 98. cum tribus
«absque fre-— duplum cumj. Cyphris -- 7.0 157 8 0180,00] Cyphris $8000,
dHone,iB — — quadratum diviforis 960 4,000000| | jquotientis primus
quo €. divi-: quadratum quotienffs - -^ ra2 $| I |&fecundus- - 3»
; w)oeo us. .: . . , anus co DÀ d ' .
| ndo dre 7 q Summa - - 9órroi7oog |
7p. — Wuédrmüm minus -- — 0.861225
funt Ad [co t . M usunnsctieenssstut dus adiibinaup igne iut 2.
ende: Cy- Refiduum tum fequentibus 15.5 7 8o 85|-
pire. | quadratum diviforis - - 9604.00 ooo] .
'quadrareinquodeatis - -- -— $241,
Summd - - - 96194843211
qpsdratum minut - - : 2 490241 .
Refiduum cum (equentibus - - - 9.40 SocO|quetientis quint
quadratum diviforis - - - 9604.00 0000|& fextus - - 48.
«quadratum quotientis - * - 2 jo4| inreper quotiens »-
O7 Sptnaequalis quadratc&o 6153.421:0304..— 357? 49-
mE . Dividen-
-
| CA P. P3 E »
| Dividendtis f 77974 Q 67183. dieifbi» gd. & Eseis ? UE
^ duplum curn z; Cyphris -1.9 4.8 15 $6.8 /og|chnz; Cy». 4bsque fra-
quadratum diviforis 622 s 21 oo o0| . | phtis7ígoo, üionc, in
' quadratum quotientis. -- - -r44|. p & dio |
"quadrans mimus 102274158279]: | "|quoriemtis ordo dus.
| quadratum minus - « ...... 167 imus& —/,M dj;
reffduum cum fequentibus - $4535 68| jfecundusm pru, ^
quadratum diviforis- - 62252100 oo| |quorientis -
quadratumquotientis - -- ir ;6| |tertius &
Summa. - - - . 6130.66 47.2.4 4| |guertus 34.
quadratum minus - - 1... 5765; H quotientis
refiduum cum fequerítibus - - 816 goo! tus&
quadratum diviforis -.-- 6 nni o doo [extus . 5$.
quadrattüm quotientis - . 31 «nteger 0S. P
'" Summazqualis quadrato 622a ^ , THOGIDS
—— de equans quadrato 6234.0409536 125456.
| Dividendus fit -: - 6790 5183loo 6. divifor 6789. & cum una, Exeorpl
duplum cum Cyphra - 13 4t 036 6l o ilz oladjécta Cyphra. 67890. pl. 4.
quadratum diviforis 4609.05 410 0 Pique [oa ^,
dosdcumquorsm - - 9óog| | jiUlimicalici - ^88 ructy dui.
4. . $utma - « 3612,47. 1070 pn Virimi diviforis ^ 7-9? nri (f djuj-
quadratum minus —— . 368144 | .[. [12 Primi quotientis -- 98 dend; una
n tefiduum cum fcquentib. I 039 2 001 et 4djicirn- .
quadratum divifaris - 4 609.0$ 310 o| - |2: Ulltimiradicis - . 66 d4 Cypbm.
quadratum quotientis - - $7746] |z, Ultimi diviforis - - oo |
| Summa ».e- 4.6 19.4.5 0 477 A. Medii quotientis -- 76
quadratum minus -- 5.377 156], |. un
refidnum cum fequentibus - - 7,3 32 120|:. lUltimiradicis - - 44
quadratum diviforis - - 4 6o 9.0 5 a1 Qojz. Ultimi diviforis - 290
Summa equalis quadrato 46 16,58 4 I : 6 quotiens integer - 987654.
. Dividen-
MED" € 4 P. F.
Bep; — Dividendusfit - - 41637 217
probationis -
é2à cirre- duplum dividendi - - $527 44 ) 4
Divifor veró $901.
;, Ultimi radicis - - 6o
?, Ultimi diviforis - - mr
fonde e- quadratum divif. 79 133 14 169
xemplo pri- quotientis quadratum -- 2209 |
JF. 1. Primi quotientis - 47
Girpd. Summa- «794-8438 39 ss pacis -. 2
wifor b4per quadr. minus - ——.683690| 477 uud
gi. nmnvetros. reiduum fequent, aut. 1 602 23 4 dirumus diviloris uL.
quadratum divi, 7923.3 14 169|Ulltimus quotientis - .9
quotientis quádratum - - - 81! quotiens 479. absque ,
Summa equalis 7924.91 6484 fradione, |
quadrato. 2
Qum
qumMMEEENMNDEMESP
Dividendus fit - - 585515[8 5j. & divilor - - 35769.
Byemgl, 8. duplum - - - i170616 371o6|2. ulrimi radicis - $y
probationis quadratum divif. 12 79.42 1361 1, Ulritni divif. - - 6g
ergà cor't- quadratum quotientis - - 156| | |; Primi quotientis 16
9». qammibunsenguu.
ph z$ 6g. summa - 712 80592244
'" i»4we& quadratum minus -- 56611;
1, Vlrimi radicis .- oj
4
| P fv ,fefiduum fequent. aut, 1601357]. |" Ultimi divi. -- à9
ve, quadratam divifor. 1279.431361| |. Medii quotient. 36
Quadratum quotientis - - 1296] |
3, Ultimi radicis -- o6
?, Ultimi divif. - - 69
2, Ultimi quotient, 5,
$umma -» - - 1282014494
quadratum minus ——-$,9980 2 5
te(iduum fequentibus auct, 2.646906
Qquadfátutn diviforis - - 1279.42 1361
! 1369
' Sumta tqaalis quad, 1 2 85.0 696561 -
. quotiens 163637. abs-
que fractione.
Dividen-
-
€ A». F. ^^. |
. Dvidendusfit - - 1681711 [^ á 6& Divifor fjz9. Even Ex
opm - - - - d presetiomp
2. Ultimi radicis 38 ergó torre-
quadratum divifor. 7i 61227 7961 2, Ultimi divif. 19 /fondens e-
quadratum quotientis - - 361 xemplo $..44
q 2. Prími quoti. IO (p.. €f ix
que divifor
qnadratum minus - —-. 546644, Ultimi radicisga, quiw-
refiduum feqnentibusaudt.t2.5 10149 Mu Uinmi divif. 19 eu, niit
quadratum diviforis - - 7 262.27 796 1| |2 Mediiquoti, 73705
quadratum quotientis - - — 5329| [2 Ultimi radicis23
Summa .- - - 7274.793439| |2. Ultimi divi. 19
quadrátum minus - - ——... 725264 , Ruenient. ultim. 4
rcfiduum fequentiaudum - - 68175 2]
quadratum diviforis - - 7262. 177061 quotiens 7.
quadratum quotientis — - - 16]
Summa zqualis quadr. - r.- 7262. 959729.
Dividcndus fit - - --3 88 88 ó [ 4 9! 9759 & Divifor, 89 7 5 p. Éxempl, p:
duplum - - - 1:17772299|95]:18/2. Llltimi radicis - - $8 ?& quo duc
summa - - 7165641745] | |
quadratum divifor. 8056.678081] — . Ultimi diviforis - - 59 nunreré pofl
quadratum quotientis - - - og oz 2. Primi quotientis - - -penüumimo
n4 rennen reum, D, f MINI 4 quae |
Summa - - 207 4.460181
quadratum minus -- —.46016 ;
refiduum cum fequentibus » - 1795 ^ m bulis
quadratum divifor. 8056.67 8081 tantummodo
«quadratum quetieatit— - - - O| |; Ultim radicis » - 6o d:frerumt. ue
Summa - -80;5.6 (7 98.6| 1. Ultimi diviforis - - 59 nizzte yel
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gefiduum cum fequentibus --- «- 217 93 1 18 m 4dbnit,
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quadratum quotientis - - 9604 2. Primi quotient, 9&
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quadratum divi: 3224.990521| | 2. Ultimi divifor. 89
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préorifomi- dupum | - - 165540é1i 5p. 1. Ultimi radicis - . 9á
PA quadratum diviforis 6983.458409 2, Ultimi diviforis - 97
quadratumquotientis - - 9801 2, Primi quotientis . 99
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refiduum cum fequeutibus - 185585 2, Ultimi diviforis
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divifore tantüm unus ad dextram poft punétum fit reliqvus,
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viforis addendo, fütminam examinando, eidemque quadra-
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diviforisebíaloti : d. quadrat quotientis: e. füutumam, |
trium additorum, némpé dupli dividendi , quadrati divffo-
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quot im diveiforc paofbqninq; numeros initiales fant reliqui.
b: quadratum uflius vel plurium numerorum qui in divifo-
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tabulet — YemiTOtootoo6O0. "usque afcendit, fed ne quis pntet, hic
quando Di-termyingtiearundem, potentiam, quomodo eadem in quo-
papi ,. Vismajore divifore peragi poffit, docendum adhuc reftat :
cr)que Et quidem in*ali cafu, abfciffis à divifore decem numeris
quàm, Operaátioinftituerda eit;per hofce 10; numeros pláné fecun-
4600000000? dum tenorém $,24.. &y.Imjus capitis "usque düm pro in-
veniendis tertio:& quarto quotientis tiumeris, ad additio-
nem quadrati quinque initiálium ad 'finiftram diviforis per
' ventum fit, antequam enim hac additio fiat, rurfus ficut in
$14. tot adjiciuntur, de duplo dividendi,numeri,quot poft
'decem initiales diviforis funt. numeri reliqui , atque fi-
militer
CAP. A E
militer horum reliqndrunr, quadratum additur dusdrato:
quotientis inventi, & ab hac fummi fübtidhitur quadratum
differentiz,quie cft mcer quotienteim invéntuni & di tos ré-^
liquos numeros, atque illud quod in hac quadratorum füb- ^ ^ -
tractione remanet, porró fübtrahendum eft; ab illo refiduo,
cui poftdecem diviforis réliqui adjedti (unt; atque tunc di-
vifionis operatio fic ad finem usque continuatur, proüt ab -
initio factum eft; atque fic quidem hic procedendi modus
sícendit usque ad diviforem 1000000000000000. ad majo-
rem verà diviforem & quousque libuerit afcendi poreft, fi.
poft quinos & quinos numeros cum reliquis perpetuó ad.
fimilitudinem reliquorum poft quinque in $. 24. & poftde-
cem in boc ipfo $. Opetatio peragatur. Literg veró Alpha
beti in duobus fequentibus Exemplis ezdem , idem etiam,
reprafentant, qnod in $. przcedenti determinatum eft ; re.
lique veró quz in dict. $. non habentur reprafentant (e-
quentia : t. quadratum poít decem divifbris numeros reli-
quorum : u. füammam hujus quadrati & quadrati quotientis
inventi : x. quadratum differentie qua eft inter quotientem
inventam & modo dictos reliquos : r. differentiam intet. . —
quadratum modó dictum & fummam modó dictam : f. re(i-
duum , fi hec differentia à refiduo i. fübtrahatur: y. reli«
quos poft decem diviforis : z. differentiam inter
hos& quotientem inventum,
Lt Exemplum
-—
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4m
CAP. f.
Bxemplem t, lsgue fsaclione. im quo Divifor babes
: duplum -- 905643
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L 4 $. 3. Pofitis
ENT | | CAP, F.
Contela mo
100000. facilis veró eft horum duorum cafuum folntio ; fi
nempé in utroquecafu non quinque initiales ceu divifür ac«
cipiantur,fed tantummodó quatuor; rcliquis omnibus ma-
. pentibus, ut in precedentibus dictum eft.
$.29. Quemadmodum jam ultró fatemur, divifionem
Demonfne- per tabulas quadratorum, in cafibus illis , in quibus divifoe
tío divifo-
mis cafibus
eonfultó o-
priffa,
qti
. major eft quám 99900. nihil aut parum compendii afferre;
rei Mathematicz, przprimis cüm à me inventa fit alia divi-
(ionis ratio, quz & huic per tabulas & aliis quibuslibet pat-
marium facilé preripere poteft, duplicató nimirum:&quin-
tuplicató per dimidiationem folütn divifore, ad. quosvis
quotientes fimplici additione & fubtractione inveniendos,
ita demonftrationem ejusmodi divifioni fübjicere haud ne-
ceffarium effe exiftimavi, fperans hic cum Cartéfio in fine;
fuz Geometrie, à pofteris mihi gratias habitum iri; non fo-
lim pro iis, quz in hoc capite tüm utilitatis, tüm curiofita-
tis gratià explicui, fed etiam pro hac demon(tratione;
quam confultó omifi, quo ipfis voluptatem il-
lam inveniendi relinquerem.
'
-
emm :
"- C 4r.
N957.292€ 2
| CAP. VL. | |
Dc Admirando Tabularum ufu in numeris figura-
tis preter quadratum, reliq uis, Trigonio, Penta-.
gonio, Héxagonio &c.in infinitum. BEEN
:S, t. |
C ex quatnor aut quinque folummodó operationibus Continua
tota coríftet Arithmetica, quz funt Additio ,. Subtra- 4 4a.
tio, Multiplisatio, Divifio & Radicum Extractio, adeo, ut
'Cartefius non veritus (it hos ipfos Arithmeticos terminos,
án fgam Geometrigm introducere; additio ,veró.& fubtra-
£Xio :&nt:.& fimpligioris :& cemmuaioris praxeos,econtrà
Multiplicatio, diy?io &radicum :extrattio altioris indagi-
nis, hzc veró, quemodo per tabulas 'expediantur, in gmace-
«dentibus pleniffimé & oftenfa & demon(trata à nobisfint
Legitimationem tétragonometriz noftre rabulariz in ge-
neralioribus & univerfali Mathefi fufficientiffimam effe, fir-
miter mihi perfuafuni eít, infüper etiam fpeciatim aliqua.
attingere haud absire effe exiftimo, attingere.dico, non ple-
né abíolvere, multó minus hic demonftrare ; quàmvis de-
rnonftrabilia fint ortmia, id quod Bata huic 'apufculo prater
intentionem.db initio tneam denegarunt,curabo tamén uc D
niverfalem Mathefin hi(ce tábulis pleniffimé applicari, & MD
demonítrari , .(i DEUS Vitam largiatur. & operam mganb : 6 ose
publicogratam.effe ffentre potuero, . primo. Yero a pro- -uM Us AES
ponam, quz circa numeros an xraverfali Mathefi i inter alia UN ^
multa proponi folent:& ad Mathefis univerfalis partem
rithmeticam potiífimum fpectant,
$.2. In Arithmeticis ex-natura ipfa numerorum y. : Bragreffa .
potius-ex naturali ipforum progreffjone, fcnobis offert a; 4rit begin
pec
6f | | CAP. PF. .
64, 4f. mnium primó progreffio vel preportio, que Arithmetica
MUPHETOEP dicitur, cujus definitionem vide in $.2. capitistertii; In qua
Fgumorem progreffione fequentia obfervarunt ingeniofi.
Numerm 6. 3. Si ab unitate incipiendo ponamus naturaliter &
sriangnla- (C, cundum ordinem numeros, continué unitate folummodó
ris femtri- ei . . u- . .
gom —— Creícentes, &tandem, ubicunque libuerit in ultimo aliquo
quid? ^ fub((tamus, atq; fümmam omnium colligamusif(tam fum.
mam efficere numerum trigonium & in Triangulum zqui-
laterum equalibus continuó difítantiis exacté redigibilem,
E. g. fifubfiftamusin numero, 7. & (ümmam omnium, 2.6. y.
4.3.1, & I.colligamus, erit ill,28.(ed hic numerus dicitur efle
triangularisquia fingulz iftius numeri. unitates , ctu pun
€, in triangulum zquilaterum exacte funt redigibiles , ita
utperpetuó uniformiterà fe invicem diítent, ut patecexape
pofitis Figuris A. & B. "
8 4 «4
duomode — $. 4: Dató itaquequovis numeró ultimó in progreffio-
4bsque 9*-. ne ejusmodi arithmetica, vel etiam numeró terminorum.
bulis iPV*-. eiusdem triangularis numerus, feu füumma omnium termi-
etum lei. norum invenitur per Regulam in Arithmeticis commu.
mumpro. ncth; quód nempe ille numerus,unitate auctus debeat mul-
grejfonis — tiplicari'cum dimidio numero terminorum, vel (i hic nu-
germinum — yyerusfit impar, ut dimidium non nifiin fracto numero ha-
V«[?V"*- beri poffit, quód dimidium ultimi numeri unitate aucti
multiplicari debeat, cum numero terminorum; Sicin prz-
Fum termi-
gedenti Exemplo, ultimus erat 7. idemque unitate auctus g,
geryt,
cujus .
|
| CAP. FN - .4
cujus dimidium 4. multiplicans numerum terminorum 7,
(funt enim in hac progreffione feptem termini) procreat —
criangularem dictum zg. Pariratione invenies, i numerus — .—
ultimus fit 48754. ejusdem numgrum triangularem feu fam- (-
mam omnium terminorum efle , 1187525745. multiplicando
nimirum ultimum terminum unitate auctum 48734. cumo.
dimidio numero terminorum 24367. ut tentanti con(tabit.
$. 5. Multó facilior & compendiofior hic eft operacio-F4cr/ier is-
per tabulas, in quibus folummodo evolvitur quadratum ul. ehula pte
tími termini eidemque additur, fua radix i.e. ipfe ultimus ^ 4 ^
terminus, eritque tunc dimidia fumma , que(itus numerus
triangularis, feu fumma amnium terminorum ; fie vides
quód ultimi termini 7. quadratum fit 49. & horum duorum
fumma 6. dimidia veró 28. numerus triangularis; Parimo-
do quadratum ultimi termini 48734. eft 2375.002756. & ho»
rum fumma 2375051490. quare dimidia 1187525745. erit que-
fitus triangularis, feufüurmma omnium terminorum, —..:^ — Nom folie»
$. 6. jpganriffime fic fingulz radices in tabulis non, 4w4dnitifod
folüm quadratos, fed etiam triangulares fuos numeros de- ^4" rr
terminant. - angulares
| . "NHMECEL PS
' $.7. Econtra ex quolibet numero triangulari velfumms ,,,,/;.
progreffionis, radix triangularis, vel ultimus terminus vel wur,
etiam numerus terminorum extrahitur, (i ipfius octuplum 2s4dotus
unitate auctum, cen quadratum, in tabulis quaratur, ejus-/ew ze£mga-
denique radicis unitate diminutz dimidium ,, (ic. numeri ^
triangularis 25. octuplum unitate auctum eft z25. cnjus ra«:
dix quadrata, unitate diminuta eff 14. cujus dimidium 7. e».
rit radix triangularis quaita ; 2 0l
$.8. Siporrà ab nnitate incipiendo ponamus numec« fede C.
rosbinarió crefcentes &fubfiftamusin 3]jquq ceu ultumo at» ang vla s .
que colligamus furiman omninutn terminorum, repraíen- /;4 Trig.
e tabit e,
,$» CAP.PL
*abit ilIa ipfa numerum quadratum, de quo, càm füprà,Ca-
pitetertió ex profeffo actum fit ; ad (equentes fimiles pet-
gendum lüceft, 2. .
Numeri -.$.9. Si itaque porró ab unitate incipiendo, ponamus
oequm- humeros ternarió crefcentas, & fübiftamus in aliquo cca
ftu quin
quangula- ultimo, atquc colligamus fummam omnium terminorum,
wu, quid? Teprzfentabitilla ipfa numerum pentagonium, fcu quin»
"quangularem ; quia in appofita figura pentagonia regulari
: ternére licet, quód ad punétum werticale, ceu unitates
"primé accedant, ad trinimurn quatuor puncta, ad forman-
"dum pentaponium primum regulare, deinde porró acce-
"dunt tribusadditis7. puncta, quz cum aliistribus punctis,
*-Fuerutit in pentagonio priori
quali diftantia, formant fecun-
Yperitagonium, fimiliter porrà
'dunt tribus additis ro. puncta,
cumaliis y. punctis, quz fue-
t in fecundo pentagonio, for-
it; inequali diffantia tertium,
tagonium , non'minus porró
. " --..edunttribus additis 13. pundta,
"quz cumaliis. pünctis;quz fuerant 'in tertio pertagonio
formarit inzqualidiftaritia quartum pentagonium ; &c. 'O-
mnia vero punétaiin tota figura C. vel alia'quacunque tma-
jeriinunam fümimám colle&ta efficere dicuntur,numerum
pentagonium, cujtisradix 'perttdpóriiaeft latus :pentagonii
maximi, ultimus vero termirius progrelfionis, juxta diffe-
"rentiam 3. repreíeritat tria maxinri peritagonii latera.
n $.10. Dató jam ultimó termino in progreffione ab:u-
A— "nitéte drithmetica cujus differentia eft j. fumma omnium,
Aly iin. Ctritünorum feu numerus pentaganius absque tabulis dre-
Wwioui
.
t.
(nt^
CAP. VL | c T D
iori via inveniri non poteft, quàm, quód tertia pars ultimi /? pentsges
termini binarió aucti, multiplicari debeat ,, eum. dimidia; "hu. daté
parte ejusdem quidem ultimi termini fed unitate folüm uu- progrefpe- :
Qi, fi modo ultimus terminus fit numetus impar : fic ex. BT- terminó.
ultimus terminus fit 5. & binarió auctus 15. hujusque tertia Exemplum '
pars j. multiplicata cum dimidia parte ultimi, termini uni- i» parvi:
tate aucti, nempe 7. producit (utmmam omnium termino- "mers,
rum velnumerum pentagonium,3;. S1 vérà ultimus tera "
minus fit numerus par, dimidius numerus. termingrum,
multiplicari debet, cum ultimo, unitate aucto; Sic ex. er. .
ultimusnumerus fit 65658. & binarió auctus 65640. cujus Exemplum,
tertia pars 21880. hujusque ditnidium 10940. multiplicatum 7/7 7*7
€um ultimo termino, unitate auclo, 65639. producit füm- ^
mam omnium terminorum, vel numerum pentagonium.,
718090660.
S. 11, Multó breviüs hac omnia unt. per tabulas, in poti -
quibus folummodo querendum eft quadratutà ultimi ter- Mibiead m
mini, ejusdemque binarià aucti tertia pars , eft addenda i- Cf imrue-.
pfius radici ceu ultimo termino, fümme dimidium, erit niazur per.
(umma omnium terminorum feu numerus pentagonius. Sic mbülas.
ex. gr. ultimi termini rj. quadratum eft 169. cui (i addatur Exemplum.
binarius 2: erit (umma 171. cujus tertia pars 57. addita radici párvi
13. facit füummam 7o. cujus dimidium 35. €t ut anteà, (umma mumeris
omnium terminorum, feu numerus Pentagonius : Similiter.
ultimi termini 65653. quadratum binerió au&tum eft 4308. r.,, esplum
3479046. cujus tertia pars1436115682, addita radici 65658. facit jj 4,5,
(ummam 143618320. cujus dimidium 718090660. e(t, ut an- majoribur,
tà fumma omnium terminorum,íeu numerus Pentagonius, !
$. 12. Datà viciffim fummáà. omnium terminorum feu; -
numeró pentagoni, fi queratur ultimus terminus in illa., D«ró sswe«
progreffione ab unitate Ma continua differentia jy. " ó pent gos
& tuna
'a
yf CAP. Pt
890, quomo- tunc fumma multiplicetnr cum numero 24. & exhoc pro-
- P ircmwe ducto unitate aucto extrahatur radix quadrata, atque abil-
pragreflonis la radice fübtrahatur numerus ternarius, j. refidui dimidi-
vermimw«. um erit ultimus, qui queritur, terminus; Sic ex. gr. fumma
(5 35. multiplicata cum 24. producit 840. cujus producti, uni-
JBxempism. tateaucti, radix quadrata eft, 29. & fi ab hac fübtraliitur, 3.
$nparvis peftat 26. cujus dimidium, rj. eft ultimus in progretfiong,
PENES. usficus ;/fimiliter (umma 7158090660. multiplicata cum 24.
Exe »rplam producit 17254175840. hujus producti unitate audi radix
in majeri. Quadrata eft 15127 9. quod ipfum facili negotió, benefició.
ém numeris, S. 10. cap, 4. experiri licet , fi ergó ab hac radice fuübtraha-.
" £ur 3. erit refidui dimidium 65638.ultimus progreffionis que-
fitus. | |
Qunomode 6.15. Si verà per datum numerum terminorum,ceu radi-
pér radisem cem numeri pentagonii, velimus quaerere in eadem pro-
geniagtin- oreffione continu& differenti 3. fünmam omnium termi-
ni pio norum feu numerum pentagonium, id absque tabulis cffi-
matar eju. Cicmus, fi quidem numerus terminorum, feu radix numeri
elem penm- pentagonii, fit impar, multiplicando dimidium tripli, uni-
gosin 45r- cate diminuti, & facti ex nnmero terminorum, cuim, eodem
que tsbnliz vwyrmero terminorum, productum enim erit qux[ita (umma
Exemplum vel numcruspentaponius; Sic ex. gr. fit numerus termino-
, 9554775. vum, feu numeri pentagonii radix ;. & triplum hujus, 15.
We*c^ — atqueunitate diminutum L4. cujus dimidium 7. multiplica-
tum cum numero terminorum 5, producit 3j. quafitam.
Tummpatn, feu nimerum pentagonium ; 5. enim puncta nu-
| merabis infigura, C, $. 9. hujus capitis. .
Sinis fi $.14. Si veró radix numeri pentagonii fit par, tunc
pc dimidium ejusdem multiplicandum eft cum triplo uni-
Axemplum. tate. diminuto , ejüsdem "adhuc radicis: Sic ex.gr. fit
numeri pentagonii radix data 218go. cujus quaritur nume-
2E rus
" 22
€AB.TVL »"
fs pentapónius : Triplum ejusdem eft 65640. atque hoc u-
mitate diminutum 65639. quod ipfutn multiplicatum cum,
dimidia tadice 10940. producit 718090660. numerum pen-
tagonium quef(itum: | |
$.1. Rurfus per tábulas hic accrefcit egregium com- Somos
pendium, nam datà pentagonii rmmeri radice, ipfe nurat- le mi »
rüs pentagonias (tati 'imvtrütuft , quxrendo radicis qua- N
dratum, eidemque addendo ejusdem quadrati dimidium,
acglectà fractione ex imparitate accedente , atque ab hae
füuma radicis dimidium fubtrahendo, rurfüs neglectà im»
paritatis fractiotie, ^ Sic ex. gr. fadicis 5. quadratum eft5. Exempíens
huic fi addas ejusdem dimidium 1absque fractione, (ümtma * Pervie-
erit 37. à qua fi (ubtrahas dimidium radicis 2. absque fra- "nts
«tione, eeftabic 55. quafitus numerus peritagonius, Simi- 4.
liter radicis 21880. quadratum eft 453: 754400. huic fi addas;, ned
ejusdem dimidium 23957100. fürmima erit 718101600. à qua bur sumerik,
fi fübtrahas dimidium radicis 10949. reftabit 713090660. .
quazfitus numerus pentagonius. i 0 7
$. 16. Dató viciffim numerà pentágonià, fi queratur Duspads
ejus dem radix, id efficiemus, (i iumerus pentagontus mul: s4meri peni
- tiplicetur cum numero za. & ex producto unitate aucto; egomiex-
extrahatur radix quadrata, hisjus radicis anitate auctae fexca tiber nili.
pars erit radix pentagonii quafita : Sicex. gr. nutnerus pen- dix penta- |
gonius 35. maltiplicatus cum 24. producit "ritate fuper- Exemplum
addita 41. cujus radix quadrata unitáte aucta eltjo. &liu-/5 er -
jus óta pars 5. que(ita radix pentagonii : Similiter hutneruspsmers.
pentagonius 718090660, rauitiplicatuscum 24. producit ü- Eyewp/uyy
nkate (aperaddità 172. 341758. 41. cujtis radix duadrata jam, i» numeris
anteà in $. I2, inventa eft 1279. quare hujusradicis unitate, t4ioribur,
Suxctz: fexta pars 21880. erit radix pentagonia quefila, —'- |
Mj $22 lee
(
"nu . CAP. FL
^ Mexapemier — & i7; Ulteriüs (iab unitateincipiendo;portamus nume
Kv fex os quaternarió crefcentes & fübfiftamus.in aliquo ceu ulti--
* geri ni^ mo, atque colligamusfümmara omnium. terminorum ,. re-
merui prafentabit illaipfà numerum Hexagonium, feu fexangula-
rem ; quia, ut apparet in appefita figura, D. pofitó pritnó,
pundtó, ceuunitate & deinde nu--
meró. 5, ab unitate per quater-
narium. diftante,. videmus: im fi--
, guraappofita quód numerus 5. ac-
. / €edensunitati, formet primum &:
E minimum: hexagogum , (i porró:
accedane o puncta &quinqne pun--
] Gis rurfasquaternarió differentia, -
qvoad eorund.numerum»hac g.pundta, cum trib. aliis prio-.
:^ ris hexagoni formant inzquali diftantia fecundum hexago-
Jo Unum, ulterius fi accedant ry puncta, &novem punctis rurfus;
m perquaternariud, atque fic in progreffionc arithmetica dif-:
. ferentia, hzc tredecim puncta , cum aliis quinque in fecun-
' dohexagono formant in equali .diftantia tertium hexago-
."o bum, non minus [i perró accedant, r7. puricta, codem mo-
.". dàdifferentia à prioxibusi. formabitur cum 7. eliis inter-.
^ ^ ' io hexagono in equali: punctorum diftantia quartum & in. .
hac figura maximum hexagonum ; Omnia veró puncta in
tota figura D. vel in alia quacunque majori in. unam fum-
fam collecta, efficere dicuntur numerumhexagonum, cu-
jusradix hexagonia eft, latus hexagoni maximi, fecundum:
puncta, que comprehendit , denominatum, ultimus verà.
terminus juxta differentiam 4. reprafentat tria maximi
. pentagonilatera. D
Quomodo — S. 1g. Dató jam quovis numero, ceu ultimoin pragrefa. .
is invenié- (onc arithuncticáab unitate, cujus differentia eft 4. fumma
: omnium,
"CAP, PI p
'inmnium terminorum íeu numerus hexagonius absqueta- 72. deti
bulis breviüs vix inveniri poteft, nifi, ut quarta pars ultimi 27^57 die :
Zerminiternarióaucti mulsiplicari debeat cum dimidia. "7^ &fe
;parte ejusdem quidem ultimi termini, fed unitate folüm au- au, able.
ti j Sic ex.gr. ulsimus terminus fit i7. & ternario auctus20.
hujasque quarta pars jy. multiplicata cum dimidia parte ul- piesplum .
1imi termini, .unitate aucti nempe 9. producit fummatn o- í» parvi,
"mnium terminorum, vel numerum hexagonium45. Simi- |
liter ; ultimus terminus .(it 13345. -& ternarió auctus 12348. &xemplum
hujusque quarta pars 3087. mukiplicata cum dimidia partem mwjori-
"ultimi termini, unitate auéti , "nempe 617;. producit fum- bue numeris,
mam omninm texnrisorum, xelnumeruim héxagonium- -
.' 19056051. MET mE |
—. $19. Adhuc breviüs illud-ipfum invenitur per xabulas, XQvomedh»
4n quibus folummodó quarendum .eft -quadrstum :ultimi Mim
1termiai,illius'ipliuserim tecnarib tauren aucti quarta pars, per. rzbulany
'li addatur fuiipáus xadici, «cen iltimo xermino , erit fum» daré rimé
mz dimidium quzfita:'Gtmmaomnium terminorum feü nu-772greffemi
merus Hexagonius ; Sicex. gr. ultimi termini 9. qvadra. mne.
"tum ternarió gucbumreít 292. cujus quarta pars73. addita ra- Pxehrpewe
dici 17,.facit.(ammam 9o. cujus:dimidiuny 45. eft, ut.antcà ^ Pero
fumma.omniumxterminorum , feu numerus 'Hexagortius: BE
Similiter ultimi:termini 13345..qvadratum ternaríó:auctum Exemplum
veft 152.399018. cujus quarta pars38099757. addita radici 1545. 6 mejore
.facit.fummam 3$riz102. eujus dimidium 19056051, eft, ut an- Pemerw.
"teà fumma.omninm terminorum , :(éü :numerus .Hexago-
-D1us, | * BENE :
$. 20. Datá viciffim 'füummá omniumterminorum, feu | ' ,, '
tnumeró Hexagonió, (i queratur ultimus terminus in lla ird |
"Progreifione ab unitatearitlimetici,continue differente 4...» poni debi
lbreviffima:inventionis via.erit, fi ex fümme termiihorum., iwrveiiaam.
feu
Ww C£. AP. PL
s]nmupro- (eu numeri Hexagonii octuplo unitateaucto extrahatur ra-
s efionis , dixquadrata, illa ipà enim binarió dimimuta erit quafitus
TP Provreffionis ultimus terminus: Sic ex.gr.ex amma termi-
Bxemplmi. o um (eu numeri Hexagonti 45.octuplo unitateaucto 361.
jn parum : aser
sumerig, Cxtracta per tabulas radix qvadrata eft 19. qva ipfa binarió
Exemplum diminuta, 17. eft qvaitus pregreffionis ultimus; Similiter
inmajoribu: ex íumma terminorum feu numeri Hexagonii 190560j1. o
symeris, — Cuplounitate aucto 152.4.48409. extracta per tabulas radix
p qvadrata e(t 1347.quz ip(a binarió diminuta 1345. eft quz-
€ z. litus progresfionis ultimus.
ne mdice $. 21. Si veró per datumnumerum terminorum, ceura-
bexagosiá. dicem numeri hexagonii velimus quarere , in eadem pro-
datá iwve-. gresfione continuz differentiae 4. íummam omnium termi-
gidturspff! horum feu numerum hexagonium, id absque tabulis efficie-
jul jg, 00$ fi duplum radicis hexagoni& unitate diminutum mul-
"- ») Ium . tiplicaveris, cum illa ipfa radice, fic ex. gr. duplum. Radicis
Jin parvis. hexagonizs. duplum unitate, diminutum , 9. rnultiplicans
Aumerw, Candem radicem 5. producit 45. quefitum numerum hexa-
Exemplum gonium: Similiter Radicis hexagoniz 3087. duplum unita-
jo meejoribot c. diminutum 617 3. multiplicans candem radicem 3 08 7.
s4"MIÓ, — aroducit 19056051. quafitum numerum hexagonium.
bula . $. 2. Non minus etiam hic tabule quadratorum praz-
er
! dice dard, TOBatiVam ratione compendii habent , in.quibus radicis
amvemiatar hexagoniz, qvadrati duplum, ipía radice fuà diminutum;
ejusdem nu- e(t numerus, qvi qvaritur hexagonius: $c ex.gr. radicis he--
.meris be-. xagonis 5. quadratum eft 35. & hujus duplum go. .& radice:
84$?"'^^ hexagonià diminatum 45. qvod ipfum eft numerus hexago-
Exemplum nius ; Similiter, radicis hexagonie 3087. quadratum eft--
pini . 9419569. & hujus duplum 19059138. & hoc radice hexagonià
Kaemplum diminutum 19056051, eft qva(itus numerus hexagonius.:
ia majori- | f
bw nsmtri, | ($35. Dató
1
6.23. Dató viciffim numeró hexagonió, fi qvératur e-.2vomode —
jusdem radix, id efficiemus, fi ex numeri hexagonii octuplo, 446 »sme^d
unitate aucto, extrahatur radix qvadrata, cujus ipfius radi- LEXAgOn |
cis unitate auctz qvarta párs eft qvafita radix hexagonia.. : jj, nai,
$ic ex gr. numeri hexagonii 45. octuplum unitate auctum 4exagonis,
eft, 56r. cujus radix qvadrata unitate auctaeít 20. hujusque Exemp/us
qvarta pars 5. radix hexagonia qvafita: Similiter numerii» parvis ..
hexagonii 19056051. octuplum unitate auctum eft 52448409. "wmerss.
cujus radix qvadrata, unitate aucta, eft 12548. hujusqve qvar- Exemplum ..
ta pars 3087. radix hexagonia qvzfita. ' | /» majori-
. e 44 NH.
$.24. Qvemadm. jam fatis conftat, qvód per certa data&
ultim" progreffionis terminus & fuia omniü terminorum & 45, o.
radix in precedentibus progreffionibus ad numeros Trigo-5,/eeuius -
niü Tetragonium, Pentagonium & Hexagonium fpectantes z» genere.
commodiífimé per tabulas qvadratorum numerorum inve-Tonfitw,
niantur, ita etiam in qvavis alia progreffione ad alios etiani
numeros polygonios ininfinitum dabiles, heptagonium, — ^
octogonium , Neagonium , Decagonium &c, tabula-
rum iftarum ufus, vulgari absqve tabulis operationi, qvam
impofterum etiam brevitatis gratià , non ampliüs determi-
fabimus, ufüi tabularum folüm nos applicantes, longe
praferendus eft; & qvoniam hoc femper ertum eft,
qvód qvivisnumerus Polygonius per certum numerum de-
nominatus, habeat ftam progreffioneim fecundum iftam,
differentiam, qve binarió minor eft qvàm ipfius denomina-
tio, ex.gr. heptagonius habet differentiam 5. Decagonius
differentiam $. &c. facilé proptereà dari poteft folutio uni»
' verfalis, juxta quam illa, qva in przcedentibus in fingulari-
büs cafibus propofita fünt, univerfaliter in qvibusvis poly-
goniis expediri poffunt. | |
23 N — $35.Dató .
CAP. F1,
Daró euju- ^^ e. 2g. Datb itaqve cujusvis numeri polygonii ultimà
vurg, Poly- terminó progreffionis ipli competentis, ipfe polygonius nu-
Ani iri rheru$ üniverfaliter invenitur fegventi modó : Ultimi tet-
P ermind, míni qvadrato unitate diminuto, atqve fic divifo per ipfam.
invenire i- differéntiam, debet addi ultimus terminus progreífionis,
Xm wwme- ünitáte auctus fümma'hujus dimidium erit numerus poly-
rinm Po099- s Sninsqvafitus. "Ex. gr. numerus Decagonius , fit ex. pro-
"- ^ —— greffionefecundum differentiam $. datus jam eft inifta pro-
.. .. qreífione aliqvis términus ceu ultimus, nempé, 39993. &
Aennplu P. qvritur furnma omnium terminorum,feu ipfe numerus de-
-.. . «*agonius,qvapropter, qvadratoe ipfius ultimi, 1599.440049.
. "unitate diminuto & tunc divifo per differentiam pro-.
*cn :greffionis 3. qvotienti 1099309006, debet addi ultimus termi- ,
| ^nus, ditate auctus 39994. & erit fumma 199970000. atqves
^— dimidiam hüjus 99985oco. qvafitusnumerus decagonius.
4Uvómodo — — '&.i6. Econtrà datà aliqvó.rrumeró ,polygonió qvóvis,
Waró qvóvis differentiam progreifionis parem habente, fi qveratur ulti-
«POUE?99 — mus (ux progreffionis terminus, per folutionem univerfa-
veniarar wt. cm hoc fit, i duplum illius numeri polygonii unitate dis
dimus pro, ininutum multiplicetur cum differentia lua atqve producto
«greffons — unitate aucto addatur qvadratum ejusdem differentiz, di-
verminm,É midi lujjs fumme radix, ejusdem "adhuc differen-
babeo: 4f- tS dimidió diminuta; efit qvafitus.ultimus terminus pro-
Jf iig gte(fionis, Ex. gr. numeri Decagonii 99935000. duplum u-
, nitate diminutum 199969999. mnltiplicatum cum differen-
tia faa it progreífione competente 8. producit 1599759993.
cui producto fi addatur 1é. qvadratum differentiz dimidiz,
fumi radix 39997. eàdem differeritià dimidià 4. diminu-
£a e(t 59997. elt qvafitus ultimus in progreffione terminus,
SP mimerut ^ 58.27. Si vérà dictus progrelflonis terminus habeat dif-
-'Ugovi — fcrentiam progreffionis imparem, tünc adliuc qvidem due
2 pium
AaMaeeonp emo
dre!
Qoo MBPS ue ]
plum numeri polygonii unitate-digmirututt tiultiplicetne &ebedt 2A
eum differentia fua progreffionoli fed jam producti hujus; fere |
unitate aucti, qvadrupla debet addi qvadratum differentia 77 ii
progreffianalis, hujusqye fumma Radix eddem differengid. re», avo-
progrefíonali. dimingta & Gic deinceps dimidiata erit qvas »odo proces
(tus ultimus. progreffionis terminus. Ex, gr, numeri-Nea, 4^9!/o/vexi
genii feu Nonangularis 281549501. duplyuta, unitate dimipn, "^ * fue,
tum 5670816ot. fi mulüpplicetnr cum differentia fua pror
grettionali 7. producit 3969567007. cujus unitats audi qva»
druplo 3587846803. addatur qvadratum differentie pror
gresliopalis.49. erit per S.1o.c.4. hujns (umma radix 126009.
&jusdemqve differentià progresfionali diminuta dimidium
630c!, ultimus qvafitus progresfionis terminus,
0 $38. Dató verá numerü terminorum jn progresfione. &,, e.
cuilibet numero polygonio competenti,(ive qvod idem, da- Zzt4 2er
«à Pedicealienjus numeri polygenii, ipfe numerus polyga- mice pely-
nius univerfali modó invenitiür, fi qvadratum radicis poly- £eriaimve-
gnis, fuá ipfius radice diminutum, tnuleiplicétut cum dif- n " s
erentia progresfionis, qvxfito polygonio numero compe- rupi, -
gentis, & producti dimidio addatur eadem radix polygo-
nia, (ic ex.gr.Radieis pelygonie fooco.qvadratum 25000000. Zxempinfs,
fuà radice diminutum 249950000.íi multiplicetur cum diffes. :
rentia progresfionali 8. producit 199966000. cuj* dimidTum,
9998009090. additud, eidem; xádici oo. füumma: eri&..—
9998)/coQ. quefitug nutherus degagonius, . 0 0.0 oer 0013
— $39. contri (ie quoxis polyganis mutnero radix ejs- Ausus .
' dem (is extrahanda, qvs.eft labiis figuoat maxime in polygo- dx 4o
Jo, numerum detum polygoninm comprehendente , & fi- Dyganió ipe
mul etiam eff &urmerué tcniminprum in.progree(ione illi nu- 7e; zrur ns,
mero polygonio competentis ef&cietur hoc, fi numerum. dix ges...
.flasum polygonium multip:es cum octuplo differentig pro» — «s ««^
Nt gresfio« |
4|
| M CAP, V1. :
.. .— éresfionalis & producto addas qvadratum differentiz, qv
— ' eftinter differentiam progresfionalem & binarium nume-
sum, hujus fümme radix qvadrata atqveaucta dict4 illa dif-
ferentió, qve eftinter differentiam progres(ionalem &nu-
* - 4mnerum binarium, s. diminuta vero, (i binarius numerus ft
major differentià progresfionali , qvod qvidem in folo nu-
: merotrigonio accidit, fi dividatur per duplum differentiz
progreslionalis , qvetiens erit radix polygonia qvafita, fic
mE er En numerus decagonius 99985000. multiplicatus cum;
differenti progresfionalis 8. octuplo 64. producit
6399.040000. cui fi addas qvadratum differentiz, 6. qva eft
inter progresfionalem differentiam $. & binarium 1. nimie
rum 36. fumm radix qvadrata 79994. aucta dictà differen-
tià 6, nempe 8oooo. fi dividatur per duplum differentiz
progresfionalis 16, qvotiens erit $000. radix decagonia qvz-
fita...
casts
CAP. FIL
De Tabularum Ulfüin aliis numeris proportionali-
bus & progreffionalibus Arithmeticis, Geo-
AMirtesdune . ráetricis & Muficis, . |
eur prace- | : $. 1.
piti » (n Vod in przcedehti capite de numeris figuratis prepo-
aritbmeticá fitum & (olutum v(t;ulteriis ad alios cafus inarithme-
progreffoni tica proportione in genere( cujus definitionem vide in $. 2,
generi. cap. j.) cjusdemqve progresfione extendi poteft, nimirum,
:Qvomedo [i in aliqva progresftone aNthmetica detur minimus-& ma-
ximui inus iffécentia progresfionis, & qvzra-
"minors o. 0f fümmaomnium terminorüm, efficietur hoc, fi differen-
onim, in. laqvadratorum majoris termini atqve minoris dividatur
grogresfrone
eritémetica; - per
P ond
-—-
CAD. P.IL. ! tp!
per differentiam progres(jonalem & huic qvetienti adda- (^) Datis,
tur füummaiRarum terminorum datorum, ut fiat fumma ,, maximó
cujus dimidium cft fammaomnium terminorum qvzfita "ét iP
Sic €x.gr. inter qvadratum 62738849. minimi termini 793. & differentia
qvadratum 3453,207696. maximi termini 58764. differentia. progreffo..
cft per fubtractionem 3452578847. qv divifa per differen- *s*.
- tiam progresfionalem 29. fit IIQgOj4.44/3. cui fi addatur Exemplum,
fumma extremorum terminorti $9577. fit (umma rr9tr4 000,
cujus dimidium 59557000.elt fumma omnium terminorum
qvalita. — TE E
$.2. Si dentur, minimus terminus, differentia progres-- Das
fionalis atqve numerus terminorum & qvaratur fumma o- () ba
mnium terminorum , in progrefsione arithmetica ; tunc sy»ó, diffe
qvadratum numeri terminorum dati, eódem numeró ter- remiá pro-
minorum diminutum, multiplica cum data differentia , & £reffemli
adde producto duplum dati minoris termini , multiplica- 9 eimerd.
tum cum numero terminorum , fumma hzc dimidia erit Mice
qvzfita fumma omnium terminorum. Ex,gr.qvadratum,
1j2.39002j. numeriterminorum dati1234 j.ebdem numeró
terminorum diminutum 152386680. maultiplicetur cum 4
data differentia 22. & producto 33522506960. adde duplum
dati minoris termini 123. nempé 2416. multiplicatum cum,
numero terminorum datoD34j.ut fiat productum 3036870
antecedenti producto addendum pro fümmá 3555$4.3830.-
cujus dimidium 167777191. et fumma omnium termino-", -
rum qvafita. - 0000. sl
$3. Si porró dentur maximus terminus , differentia. G) Bwtár
pregresfionis atqve numerus terminorum .& qvaratur "x2 —
fumma omnium terminorum; A duplo maximi termini P diferent 4
differentià.datà aucto & multiplicato cum.numero termi- 55...
norum; (ubtrahendum eft qvadratum numgri terminorum ss arg, ww-
- N3 ^ ^ dati
—-
a. CA». Pr
meró tevimi- dati, cum differentia data multiplicatutt, refidui di midi uie
wóru. —— erpjt qvafita fumma omn:um tertüinorub. Ex. gr.à miaxi-
Kxemplunt. fni tetmirii 271691. duplo ;j45n. differentia datà22, andtà
$4340 & multiplicato, cutn numero terminotumni 12345: ni-
mirum 6708112380. fübtrahatur ; qvadratum numeri cermi-
, norum dati 152, 399o:j. multiplicatum cum differentia date
"oH. nintitum 335277 8550.refidui3355543 816. dimidium
uemod 1677771915. eft fümma omnium termitiórum qvafita,
Svenietur. '. &. 4. Si dentür alicujus. 'progteffi ónis arithmeticz füm-
maxim — ma omniumterminorum, nec non.progreffionis differen-
6 4 LM $- , tía & minitnus terminus, & qv&ratur maximus terminus;
Be, grub turic illa data differentia aüt cft par aut impar, (i par fit,
vU ca tero c
ses multiplicetür duplum fumme data, rhinimó terminó dimi-
(j)B«ne Wutum,cum differentia progretfi onis & produdto addatur,
máe- qvadratum dati minimi termini, nec non qvadratum dimi-
gniam (67- die differentke ; hujus fumriz radix dimidii datà differentià
minoris diminuga erit qvafi tus maxiinus progreffi: ohis terininus,
Mei po ^ Ex.gr. fütumz dat 6348000. düplum 12696000, minim ter-
minó G dif-
ferentié, minó 354. diminutum 1695646. multiplicetur cum diffe-
rentia progreffi onis n. & producto 132347752. addatur. n2516.
Sj diferen- radratum dicti minimi termini 354; nec rion 36. vadratum
tía fit na- d i d.
perut par. imidle differentia 6. hujus fümmz 152473164. radix qvadra-
Exemplum, ! fa 1348. dimidia differentià 6. diminuta 1342. eft qvati-
etus maximus | progreffinis terminus, ..
Si dferen--— $$. Siveróin przcedenti cafu differentia progrelfio: io-
pia fi "M nis fit impar, tunc adhuc qvidem mulriplicetur düplüm
meri fümmz datz' minimotermino diminütum, cum differentia
gare progreffionis & producto addatur qvadratum dati mininri
. termini, fell jam hujus fumme qvadruplo addi debet qva-
: dratum differeütit progréffionalis, ut fist fgmma; "habens
0? radicom qvadtatam, cüjys differenti" progresfi ondli dimi-
nuta
Moa A
CAP.PFIL —— f^
nut dimidium e(t qvafitus-maximus progreffionis terni-
us, Ex. gr. fümmz date 59557600. düplüm n9ma4ooo.miiti-
2mó terminó 793. diminutum 911207. multplicetür cuai, Exemplum.
differentia pro gres(ionis 29, & producto .3454233003. addá-
tur 618849. qvadratum di&ti rhinimi termini 793.hujus fum-
7m2.345491182- qvadruplo:1819647408. addatur qvadratum
ifferenti ptogresfidnalis 841. ut fiat furnma 15819648249.
habens-radicem, juxta $. 10. capitis 4. inveniendam, 117557-. .— 5.0.»
cujus differentià, progresfionali 29. diminutz, dimidium — . ; 4
58764. et qyalitus maximus progresfienisterminus. . —DEEELL
: à , to ( «
i
|. $. 6, Si detur alicujus progresfionis arithmetica fum. (2) Daiir
maomnjum cerminoruam, nec non differentia progresfionis /uppp 4 o-
& numerus rerninoeum, atqve qvzratur maximus téril- zmium met
nus ; tunc duplo fümmz addendum e(t qvadrátum.numeri "rerum
terminorum, ,epdem. pnumeró terminorum dirinutumy & ffo
*ic multiplicatum;cum differentia data, additornm horurfr, , DE op.
fütuma divifa per duplum. numeri terminorum, elt quzll- weró rez-
tus maximus terminus, Ex.gr. duplo fumma*67777191 i nil- simorwnr.
mirum 35555$jo.addatur gxadratum152.399035. datiüumé- — Qvome-
E terminorum 2345, diminutu m hàc ipío nnmeró térmiro- do imrvenia-
qum, nempe 153386680. atqve fic multiplicatum cum data", niii
«lifferentiazz. nimitntn 3352506960, horum additorüm füti- ,,, gne fo
3a 679805079. divifa per duplum numeri terminorum nis AViLbtni-
24690. nimirum.371691eft qvalitus maximus terminus. . eie. ' 7
^0 78.7. Similiter fi detur alicujus progresfionis arithtné-./,3 sias
ic? lumma. omnium terminorum, .nec non progres(iofíls /ywené wew-
differentia, numeri paris atqve maximus terminus, & qva-viseramsy;.
Tatur minimus terminus, tunc-À qvadrato maximi dati ter- difereneé
mini, dimidió differentiz date aucti, fübtrahendum eft pro- 7, "PAM
«luctutn multiplicati dupli fammie date, cum dta differef- sj 4,57
"tig,refiduum habebit radicemyqva dimidió differentiie dàta e« fitnume-
' Ie HUBA er ve Un 7*7 ends "pars
"0j
|. "4 G A P. VTL. D.
Wxtnplutr, aucta, eft minimus progresfionisterminus qvzfitus. Ex.gr.
| 5 qvadrato 152.473104. maximi dati termini 342. data diffe-
rentiz dimidió 6. aucti; fübtrahatur productum 152352000.
factum ex multiplicatione dupli fütnmz datz cum data dif-
ferentia, refidui hujus r04. radix qvadrata 348. dimidio
differentiz auda, 354. erit minimus qvaitus progresfionis
tefminus. : mE tacto
Si aferen- $. 8. Si veró in precedenti cafu progresfionis differen-
ipi tia fit numerus impar, tunc addenda eft hzc differentia du«
pe. |, po maximi termini data, atqve ab hujus füummz qvadrato,
/— fubtrahendum eft octuplum ejusdem differentiz progres-
^ .fionalis, multiplicatum cum data füumma omnium termi-
norum, refidui radix, aucta differentià progresfionali, &
fic dimidiata eft qvzfitus minimus terminus, Ex. gr. diffe-
Bsemple? centia 19. addita duplo 167528. dati maximi termini 58764.
facit fummam, 117557. à cujus qvadrato, inveniendo per $.;.
'" €ap.4- 1819648249. fübtrahendum eft octuplum differentiz
231. multiplicatum cum data fumma omnium terminorum
(c $9557006. nempé productum 13817224. ut fubtractiene fact .
- remaneat 2.424249. cujus radix eíl 1,57. & aucta differenti
progresfioriali 1/86. cujus dimidium, 79;. eft qvafitus tmini-
mus terminus progresfionis, à
(3)D4tis—— 6.9. Siporró dentur alicujus progresfionis arithmeticz
femmás- fümma omnium terminorum , differentia progres(ionalis,
niam te atqve numerus terminorum , & qvaratur minimus pro-
nec non dif. gres(ionis terminus; tunc qvadratum numeri dati termi-
ferentió f. norum, diminutum eódem numeró terminorum atqve fic
mumeré ter- multiplicatum cum data differentia, fübtrahendum eft 4
minorem. duplo date fummz , atqve hos refiduum divifum per du-
gxemplum. plum numeri terminorum,erit qvafitus minimus terminus.
Ex.gr. numeri terminorum dati 1345. qvadratum 152399025.
J E . códem
P da c9
.
.
I e *
MENMESMP CAP. Y 0 0, 77 0a
eódemnutteró terminorum diminutum 1/3386680. atqve, *
fic: cutn data. differentia 22. multiplicatum , 3352506960.
fübtrahendum eft. date (ümmz r6 77771915. duplo —
3355543830. refiduum 3036870. divifüm per duplum nume-
n terminorum 24:690. nimirum 125. eft qvafitus minimus
rminpus. mE | B
8. 10. Potró, "fi dentur fümma omnium terminorum, Qvomode
nec nonminimus & maximus terminus progreffionis arith« svesiatur .
metice & qvzratur differentia progreffionalis , fit hoc fe-2'eg*effe-
qventi modo: «A qvadrato maximi termini fübtrahatur milis diffe -
qvadratum minimi termini &fe(iduum dividatur per diffe. "^'^.
rentiam, qvz eft inter duplum fümmz datz, & fgmmam, / D«ri
- fnaximi & minimi termini ex. gr. à maximi termini 38764 má to
qvadrato, 34532076906. fubtrahatur qvadratum 628849. tni- minorum ,
nimi dati termini79;. atqve refiduum 3452578847. dividatur 44, minim
;, per differentiam 119054443 qvz eft inter duplum, fümimz maximé '
.. dátz 59557000. & fummam maximi &minimi termini $9557. Eremo E 2.
qvotiens 25. eft qvafita differentia progreffionalisc - OMBRA
^o Sari dentur füummaomnium terminorum, minimus '
, terminus, nec pan terminorum numerus & qvzratur diffe. í 2) Dti
rentia progresfionalis; tunc à fumma terminorum data priu pup.
fübtrahi debét minor terminus datus cum numero termi- sisorum,
norum dato multiplicatus, hujus re(idui duplum fi divida- "r/mó ser-
tur per qvadratum dati numeriterminorum, eódem nume- mod & vits
ró terminorum diminutum, qvotiens erit qvafita progres- porum, Pls
fionalis differentia,, Ex. gr. à fumma terminorum data -- Exempli "
16771719)$ fabtrahatur miner terminusdatusi23. multis —— P ek
plicatis tamen cum njmtro terminorum dato 1234. ni- -
- mirum $1843. hujus re(idui 1676253480. duplum,
3334596960, divifum per qvadratum dati numeri rermi-
UU (Um Q '" morum
—
"i
7 Y
T Cute. 4^I5
. norum diminutinm 152386680. qvoritu 35, efiqnefita pro
gtésfionalis differentia, —^ | t
t») Dara $. D: Si dentur. (umma omnium terminorgm , msaxi-
amm49- — imusterminus & numerus terminorum, &c. qveratur diffe:
rk, ' rehtiapropresfionalis, tuncdurmma tenrinorum data füb-
maximi ry. Vrahatur à maximo termino;multiplicato cammumerotcr-
: ximo término dato 271691. multiplicato cum numero tet»
venplo &.,, minorum dato 1345. nimirum 3334023395. te(tabit 1676253486.
bujit capi cüjus duplum 3552y0695o. divifum per1536630. qvod ef
m", 'qvadratumnumeri terminorum *eàdem numeró termino-
düuomed? rum diminutum, qvoriens zz. eft qvata dilfereuda pro-
ssvemiatw' gre(lionalis. —
unthute—— $ 3, Si deritut fumma omnium terminorum,maximus
. Dara tetminus & differentia in progre(fione aliqva arithmetica,
MA ni*. & qvaratur numerus terminorum; tunc differenciz datz
wig» ep. 0Ctuplam, multiplicatum tum data imma omnium termi-
— cwimerem, norum fubtrehi debet àqvadrato, qvodft vx duplo maxi-
waximórer- mi termini datà differermà aucto , refidui radix/qvadrata ,-
minó 8 dif- (ubrractaà dicto-duplo maximi termini datà differenrià au-.
fer rfe, ^ &to atqve(ic divifa per duplum -differentiz date, qvotiens
ps lieife " erit qvefitus cerminorum mumerus, modó in divifiore ni-
vides; ru, DÀ reimanferit, five divifio absqve fractione peragi potu--
sit pari fé. Crit, ve divifor dividendi parsexacté extiterit, Ex. pr. ma-
*camdió Exc), xXimi termini dati38764. duplumary2g.datà differentiá 29,
M:efs gH. auctum 117 f 7. habet tjvadratum 1819633249. fecundum
"xem. fo "S. To.cap. 4. inveniendum, 3 qvofi fübtrahacur, differentie
eréponiess: «daga Gtuplum 25. wulüplicatumcunr data fumma oni
. 4^. . - ^ ^ «m
-
N
€
e
mE . eA TIT [E
um ternmühoruis, y95y7000. nompà 198932 4606. celinzcim Exem-
qviur; 2.4242 49. cujusitadix qvadreta t9. (übrratta Adi 3$...
&o dnplo differenti aucto L77j77: relinqvit tr6000. qvod 4/9 ePi
divifüm per duplum differenti 58. facitqvotientem2000.
qvzfirum numeramtermineram, - . os ani
€. 14. Si veró.in przcedenticafti duplum differentiz dai; 47;
tz:non cexacté dividat inventam juxta precedentia radicemvidendi - '
. fabtractamáà duplo maximi termini differeritià auctó, five fi /veric par-
in divifione emergat fractio. adhzrens, qvod qvidem ra AF
risfimé contingit , tunc eadem radix inventa dicto duplo de deri.
differentià auctó addi debet, atqvefic per duplum differen, pr, ——
te dividi, ut qvotiefs.lit. qvaffitus numerus terminorum... |
Ex. gr. maximi termini dati 4997. duplum 9994. datà difz Exemplum.
ferentià 5. auctum 9999. habet qvadratum, 99.980001. t |
qvo fi(übrtabatur differentiz date octuplum 40. multipli-
— catum Cum data fümma Ottinium terminorum 24.299500.
nempe, 9098DOo60. relinqvitur I; cujus radix qvadrata r.
'addita dicto duplo maximi termini, differentià progresfio.
tali aucto, facit (ummam 1ooco. qvie divifa per 10. duplum
differentiz eft 1000. numerus terminornm qvafitus,
$. Ij. Si dentur fumma ómnium terminorum, minimus c, p jc
terminus atqve differentia prógresfionis alicujus arithmeti- fio ee. L.
€2 & qvaratur numerus terminorum ; tunc differentia datae mim rer» 5
octüplum multiplicatumcum data umma: omnium termi- rrnorsym,,
norum, addi debet qvadrato differentiz, qvx. eft inter du- miné 6$ df.
plem minimi terininidati & differentiam progres(ionalem 5,6 Pape,
datam, (umttitex hac additione fatta labebit radicem qvae ereffonali,.
dtatam; qvo (i fübtrahaeur dicta differentia, qvz eft inter 3i duplam.
duplum minoris termini & differentiam progresfionalem., 9NÓnrse reve
k re(iduum dividatur per duplum diffrentiz progreslona- mH AN
lis quotiqxs exit, ipfe qva&ieusnumerus ccrminortimi, mO* qoos ifo
Ds Oz. - lO rentia pro^ ——-
| grefionalit.
oat c
. wwum Exem-
poo 0 CÓMkRPFIE MEM
. ^ . dóduplumminimitermini dáni majus fucrit , differentia
Exemp.eor-. progresfionali datà. Ex. gr. differentiz data 12, actuplum;
rdP4"4* Q6. multiplicatum cum data fümta omnium terminorum
glo $. 4. bn- 6348000. producit 609408000. qvod.addi debet qvadra-
ju capire, tO 484-416. nimitum differentiz 696. qve eft inter mini-
. mi termini dati, 354. duplum 708. & dictam differentiam,
progrestionalem, fumma hujus 609. 8924416. tadix qvadra-
ta eft 24.696. à qva fi fübtrahatur dicta differentia inter
duplsm minoris & progresfionalem & re(iduum 24000. .
dividatur per 24. id eft dupluin differentiz date , qvotiens
efít1oo0. qvafitusgumerus terminorum, — , -
Si duplum $. I6. Si veró in precedenti cafu duplum mitümi.ter-
minimiter- mini dacifit minus, qvàm differentia prezrcesfionalis data,
miri fati? tunc eidem radici, differentia, qvz eft inter duplum mino-
oim dif. 5 termini & differentiam progresfionalem addi debet, hu-
quam diffe. TOP TAM dU
rentia pro- jusadditionis poftmodum fümma factas: (i dividatur. adhuc
grefronali, ctiam per duplum differentig progres(ienalis,qvotiens erit
£xempl.cor. qYlitus numerus terminorum. Ex, gr. differenrim .datz *
rtfpondens octuplum, 40. multiplicatum, cum data (ümmà otnniuma
e Brem terminorum 2499500. producit 9299800003 qvod addi
Mujus eapi-- debet qvadratoj. nimirum differentia, 2. qx: eít inter mi-
ys, ., nimiterminidati2.duplum, q. & dictam diljerentiata pro»
gres(ionalem 5. fumma hujus 9998ooot. radix qvadratcà cf
9999. huic radici dicta differentia 4. qve eft inter düplum
minimi & differentiam progresfionalem addi debet, (nm-
3a 10000, dividi per duplum differentia pregiresfienalis Yo,
, Qvotiensrooo. eft qvze(itus numerus serminerum;. si
JPrepomendo- — $.17. Omnes jam progres(ionis asithritéticz cafus, qvi |
vam im pro- dari pofIunt circainventionem terminorum entremoram,
pin ** differentiz progres(ionalis, numeri torminorum & füummz
AIPNIHRSA* Omnium terminorum, & propofiti & foluti fant, pracer il-
7 é 7 ; . . MD los
-d
*
CAP EIL OS. 0 0 ww
Yos qvi pritet multiplicationern & divifionem tabulariam. niperim.
: noftram decoetero de compendiis tetrágónonietriz tabu- e.t ee
. arie dil'ampliüiparriéipart ; dvóniam veró paucis etiam
illi cafus expediti po&ürit, ethec progresfionis arichmeticae
Tnatéria; qvie & eabulis'& nóftris, & cubicis, & legaritlitrti- — .——— '
cis & aliis qvibuslibér fü largitur fundamenta , plenitis per |
tracer , nec qvicqvàm à "euriofis hic:defi derari. posfity
preprimis fi alia exempla fibi ipfis exercitii grati proponc-
7t délidereüt, cafüs illos brevisfité& proponani &folvam,
sb exemplorum per fe facillima applicatione abfinendo. '
6:8. Si itaqve dértur tiinimus tertrünus, differentia & Peri
termini N-
numerus terminorum, makimisiRVéniturtérminus, finu...
3hcrusterminorum'unitatéditninttus &nultipkcetur cum.
differentia & producto addatüt mifiirnds tdriinus. ^ ^. | |
6.19. Si déritur riakttiusfermitnus , differentia & nu- Inventio C.
Tnerms teemitorum, mifirus invétiitur terminus, fi nume- minimi.
für terminorum trihiplicetür cem. diffesentia produ- ^ cows
€tum fübtrahacurá thiximó teiffálitó; differetcid'auckó, ..
!—— $.20. Si dentur rnaximus (d riiniriastermirids, nec non y Dreenriodi |
. [r*-
termifgorum numerus, differetitia' iuvenitur, (i minifnus grcfimali
terminus fubtrahatur a maximo, & refiduum dividatur per |
| riiiérum terminorunt, unite diphánutum. iti lc " |
dr Sidehtur tíbfgmiel&caninimubverminus qoc: noa. ss Vies
differentia, invenitütiRüm&erus ténininórur, fimimimus err ,,, iu
3minusfübtrahatur/à maxitro-differentià progreffionali aii- Iu
«6 lercfiduura dividatur per candem digeronciam progeee- EN
(idüalem. | o7 0019 2 * 0015989 uo tt usus
(OP a Sl dentór ovxjntur'e rims tePÁniras BetndH poesie.
«erminorumtnümerus, invenitür fümfna-omniunitermiino- farbnna. to:
sur yf dimitlia-fürmia mittiinl V máxivii fmíltiplioérur trnseter-
«um: nuiero jorninefutn; módo ilta furia (iz qumerus "7707
die. od A 03 : pars v
b
-
' y
Q7
$3 7 c'*
3
1.
T
uu
Ne C2 P. YT. |
par, baro fiámpar fic; tota illa fumma smltiplicari debet cuam
' . ' ' démidiotermiaorum numiero. — . .
praesit, S33, Si dentur minimus cermings, terminorunr nume
- ermini — vus & (umma omnium terminorum, maximus. invenitug
wAUuP! — terminus, (fngmerusterminorujgp eultiplicetur cum 4uini«
«lie. mo termino & producbum: fuübtrahatur.& duplo fummz o.
muium.tertpinerzur, aque tefidunm dividatur per. eysdem.
numerumterminorum,. .
Mewentiomi- — 6.14. Si dentur maximus terminus, numerug'tetmino-
wii tTPIDE eom & (umma óomnium- terminorum ,. minimus inyenitug-
4a. terminus, fi maximus terminus multiphcetur cüm nutnero-
terminorum, & productum (uburabatur à duplo fummz ox
mniom. terminprum , atqve. refiduum. dividatur, pes cune
dem terminorum numerum. .
Brpentiopu- ^ 5. 25.. Tandemídentur, maximus & minimus termi--
meri termi- nus, nec non famma omniumterminorum , numerus tere
ferum alia, minorunt invenitür, fi duplum. fpmms omnium tertnigo-
rum, dividatur per fummam maximi & minimi termini, ,
. 26. In proportionevelprogroffione geometrica qvid. .
| poffit noftra Tetraganometria: tabulario jam parüm. Perqni
Je pragret. ($ 27. Miffis ilüsinpropreffionearithmnetise qvàm plur
fone Geo- — simis; pauca qvedam cirea. progre(fipaam peomerricam
Amtricá I8- proponam, ubiftatim Tabylz qvadratorum vel ob hunc fag -
fu uas e 1m ufum proportionaliiim Bremertsicorum , maximi, (unt
qui? momenti, nibil énijo ferf in-tota Maghdicrebriüs o«eprrit,
inv enrieme- qvàm inventio medii proportionalis , qvod ipfum.«alcu[é
di propar- alitey deteresinari non poteft, eifi ut desa extiyema faeum., —
. BioBeln Geo- multiplicentur, & ex predutte;radix qvadrata-exrabatur, ^
Mrs s quàm facilisqueem (it extradlio radicis per ralulas fupecits.
" Sspieqvatto abunde demppftratum ei. Biogt.inter q,& 9.
! i2 , dqmedius
00 05 0 Miu PP. (^ "m
*$nedius proportionnalis ef, qvia qvater novem funt 36, cu- a, Exempla
jos radix qvadrata 6. eft qfefitus medius proportionalis ; Si- 79 Parvi
tniliterinter duos numeros 39601: & 29929. medius propor- "4 eribus
tionalis.eft 54427. ita, ut fe babeut 39601. ad 34427. ficut fe; Miti
habet 34427. ad'29929: vel vice versà, 29929. fe habet ad
34417. ficut fe habet 44427. ad 39601. qvia extremi dati.nu-
merifecummuhtplicati, producunt 1185213329. :cujus radix
. qvadrataveftütfüprà 34427. intabulis. —— :
«4.28. Porró cnam-certii :proporüionalis inventio per Z»vzm?o —
tabulas qvadraeorany fecilljta eft :qvid fecundi termini 77776 zer
«qvadracum divi(ünv per primam terminem dft qve(ituster- " Li ie
tius termiuus.. Ex, gr. fecundi itermini-dati,6. qyadratum, ri d
eft )6. qvod-per.cermiram primuni 4. divi(um!eficit.qvo-. , Exempla.
xientem9, qvzfiram rertium cerminum , (fimiliter fecundi ;, 5, eg
termini, 23427. qvadratumieft n8y.2i81ó.4jvod divifüm:per »sajoribue-
" fprimum rerminum 396ot. efficit qvotieritem 29929. qvifi- sum] o
tum*«tftiummtérmimoum. : c 00 S Me n [1 ME
Po w$.29. Sic facilé continue progreffie^geottieetica fieri Triampre»
-poteltin iifinitum, (i tertió:propértiondfi-ineenró: porrá Dorrindtie
inveniatur qvartus, pef fetuhdufsi &itertian, fie qvart& 777 0
invent) deinceps qviütue inVefiiatüry per«cortium Rc qvar- cos riw,
tum & fic deinceps, femper qvarendo dudbus prime át- quomqgue '
^ vecedentibustertium .proportionklem-tvousqve Tilemersé, dibuerir. — -
' Afurntm veróomnitm ceéminordudconipemdlese do venite -Qomod»
licet, fi dividas (ecündumi terminum per primum , ut'ha. gmveniatum —
'beasfic denominatorem proportionis ; &oc:elt, (fecondum., mena o--
qvam rationem ex. gt. dupldrirtriplam-&c. vel.aliam qvae 5777. "FP
. , lemcü mve;progreffiofacta Tits dotiominatore ficinvento, 5,55. fionis
potrà dividerida eft differentia iniasimunt& miris gemere,
progrerfióeis teHitiitiura, perit denómitiasorer uri
tte dininotütn; qvotiensanaximp:teriuinibauctus weftqva-
ZEE 2 fita
x
^-
L 00 €4P. FH MEME
- fita fumma omnium terminorum; qvod qvidem hic tradi*
tür, non, demonf(trandi alicujus fingularis compendii gra-
' tià) qvod cabulis qvadratornm prater divifionez tabulari;
am accrefcat, fed invidiz; declinandz cgusáà; & ne geome-
trica.progrefio conqveri poffit; ipfi denegatum efle, qvod
| . erithigeticz conceffum fuiffet. J H
Proklematis .— S 30: Ex pla vero fumma geometricé progreffionalium
promireculo numerorum colligit Clavius fingulargm corundem affe .
Clevi «d. ctionem, regul& VII. proportionalifatig Geometric, in Eu-
sagi 44-— clidis.fui Libr, V. def; 4. pag. mihi 416. qvam incredibilem,
jure *^** confidératione dignam, imó miraculi qvodammodó effe,
amos .. ibidem depradicat, circa qvàm affectionem plura miracu-
lofiora deducere in gratiam tabularum noftrarum liic pla--
. £ct. ej TT n
PAEED $.31. Sidetur itaqyc denominatio progreffionis feu pro-
iium rer-. portio (i.c.qvàm multiplex fit terminus confeqvens, fui ane
minorum, tecedentis vel qvota fit ejusdem pars vel qvot conftituat c-
exsreimorim jusdem: partes , de qvibus- vide. definitiones 3:4. & j. VIL
b rec Euclidis:) nec non terminorum numerus, & qvzrantur ma»
/——— uio quo. "imus & minimus termings,qyi fegum invicem multiplica-
modo aqua-ti Cfficiant, fimil; (ummats, omnium. terminorum ; ad hoc
li ejf? po(ft ? problema. folvendum Clavius d. 1, prafcribit iter longiffis
'" | . mum,nos verófeqvensyia multà hrevior,rüm ad digniora
-« aritbmeriet confiderationc,rüm ap; egregias ,CofBsarum qvantitatum.
M nid ov. Propriesates deduset,adeó up nullus dubitem, qvàm pluri-
dalonala "^. max hoc fonte hauriri poffe, eroum, de qvibus integrum,
peso, ^ opuscon[ripfit Ismael BulligJdus fub titulo operisnovi A-
' rithmetice infipitorum, Libuis (£x coraprehen(i, cujns Au .
. fhorisfaciunt inenrippem A £t3 eguditorup. (ji Anno di
pag. 108. gvanqyam ipfpm. Authorem. infpisexg; hactenus.
Rühi. nem icu 5 Selntionij veró. mee, fundamen-
La 5 tum
«4
CAP. pr D I eut p
tüm pono, numerum Trigonium feu triangularétn, per. & Solsriene
5. cap. VI. iaveniendutn. ( quemadmodum etiam dict, Bul-/*mdemes-
lialdus citató locó Triangularium numerorum ulum per-, p» e
petuum deprazdicat ] & qvantitates cosficas continuá mul-,,, e ki Am-
tiplicatione crefcehtes. (itates coff-
6.3. Duplum enim illiusnurmeritriangularis, feu tri-, 4 -
gonii cuj" radix trigonia eft data proportio,unitate auctum, Seíseio ipfa
Grit qvafitus maximus terminus trium proportionalium,( Si Ires
qvorum fumma eft eqvalis producto ex multiplicatione, dene
maximi termini cum minimo, qvi femper & univerfaliter " 2
ratione plurium terminorum; erit inventus maximus per
qvadratum datz proportionis divifus : Idem erit pro maxi-.
mó terminó inveniendo,fi Cubusunitatediminutusdivida- ^. . |
tur per proportionem unitate diminutam, | 0 .
$8.3; Duplum veró ejusmodi numeri trigonii, cujus ra-(2) Si ava« . .
| Gix trigonia eft data proportio, auctum unitate & infu-twor rermini
per Cubó datz proportionis, vel, qvod idem, qvadrató 4^". —
cum ipía proportione multiplicató , erit qvafitus ma. C^ qvi
ximus cerminus qvatuor proportionalium , qvorum mini-.
musfit rurfüsdividendo maximum per qvadratumpropor- — cc «7i
tionis data, & qvorum fümma fecundum conditionem re- |
qvifitam eftaqvalis producto multiplicatorum fecum ex-. C.
tremorum terminorum ; Idem maximus terminus invenie- EE
tur etiam fi dictum duplum trigonii numeri unitate ag- )
&um, mulriplicetur cum data proportione & producto ad-
datur unitas ; Adhuc etiam idemmaximusterminus inve- — .
nietur, fi date proportionis qvadrati qvadratum [ id eft; Zes Zesfue
Zerifi Zenfüs,]unitate diminutum;dividatur per proportio- 4*4 ?
nem datam & unitate diminutam. | EM
|. $. 34. ldem adhuc duplum numeri trigonii, auctum u- C) poil |
nitate & infuper Cubà proportionis date, ncc non qvadrati 2,,,,,..
P qvadratà |
Y
*
qvadratà five Zenfi Zensó ejusdem proportionis,erit qvafi
tus rnaxitnus terminus qvinqve proportionalium ejusdem,
tenoris, refpedu minimi termihi & füummz : Idem ctiam
maximus tertinus qninqve proportionalium invenitur, fi
dictum duplumtrigonii numeri ünitate auctum mültipli-
cetur cum qvadrato proportionisdatz & producto addatur
" eademproportio unitateaucta, Vel etiam (1à dicti dupli
numeri trigonii unitate aucti qvadrato fubtrahatur idem
. dem duplum, cum proportione data & unitate aucta mul^
Surdfol. tiplicatum ; Vel etiam, fi data proportionis fürdefolidum;
dium qvid ? (vod eft qvadrati qvadratum cum radice , vel etiam qva-
| atum cum cubo multiplicatum,) unitate diminutum di-
Vidatur per eandem proportionem unitate diminutam.
(4)8fx ^ S$35.Similiterduplum numeri trigonii, cujus radix tri-
tt?mini dem. ponia eft data proportio, fi augeatur unitate, Cubó , qva-
tuf. drati qvadrató, five Zenfi Zensó, nec non furdef(olidó, illud
] ipfum fic auctum, erit qva(itus maximus terminus fex pro-
. pordionalium,minimum terminum habentium, maximum.
fusdeso(cdur« perqvadratum proportionis dat divifüm & cum maximo
| multiplicatum,fümmam omnium terminorum efficientem;
Sepofitis jam aliis multi$ & variis modis eundem invenien-
Q»abtti/— dimaximum tertninum ejusdemtenoris,tantummodoó illum
c ubi fret modum adhuc fubjiciam, qvó preportionis date qvadrati:
du a ^ Cubusfive Cubi qvadratum unitate diminutum & fic divi-
qvid? frm per eandem proportionem unitate diminutam, eft i-
dem qvz(itus máximus tef minus, | |
() Siplees ^ — $.56. In promptu jam ex pracedentibus e(t univerfa-
4 vottunqur lis problematis duplex folutio, in qva fimil refulgent prz-
vem" cermultes aliasytres pracipux mirabiles coflicarum qvan-
citatum omnium proprietates Qvia enim duplum numeri
. &rigonii nihilaliud eft ;. qvàm qvadratum radicis trigonie
: 07 » . eàddem
"A
- 4 — aS" 3
€ A». r1 : o7 . Boc m 2L
eàdem adhuc radiceauctum, & in precedentibus folutioni- — ^ :
bus femper acceffit, (1) uhitas in tribus proportionalibus, Tr*/prack:
1) cubus proportionis date in qvatuor progreffionalibug, P*« Cefice-
(3) Zenfi Zenfüs ejusdem proportionis, in qvingye progres-, dude
fionalibus, (4)furdefolidum, in fex progrefüionalibus, fimj- oreet "e
]i etiam modó,fi porró fic accedat Zenfi - Cubus vel Cubi-/7) masia
' Zenfüs efficietur maximus terminus infeptem progreffio- tw rermi-
nis geometrice terminis & fi deinceps accedat bifurdefoli- "s /écws-
dum ( qvód eft, fi Cubi-Zenfus cum radice, vel zenfus cum ^P £F oble-
fürdefolido, vel deniqve Cubus cum zenfi Zenfo multiplj- a uu
cetur: ) maximusocto terminorum & fic porró per acces- i Coffcis
fionem zenfi-Zenfi-Zenfus, (i.e. (i Zenfi-Zenfus, cum Zenfo colecirvi.
multiplicetur:) maximus novem terminorum & deinceps Bi/urde fo-
per acceffionem Cubi-Cubi, (i.e. Cubi cum Cubo multipli- dem 4vid?
cati ) naximus decem terminorum, porró adhuc (i accedat Zes/- Zenfs
: furdefolidi Zenfus, (i.e.furdefolidum cum Zenío multipli- Zmfe
catum) maximus undecim terminorum efficietur, & fjc qvia?
:porró continuari poterit in feqventibus, trifürdefoli- Cuhi-Cubws
do, Zenfi-Zenfi - Cubo , qvadri furdefolido , Bifurdefolidi 7v!
Zenfío, fardefolidi Cubo & Zenfi-Zenfi-Zenfi-Zenfo &c. (in- Surdefalidi
gulis per contingam multiplicationem cum radice qvous- m
:qve libuerit creícentijbus, 7
6. 37. Altera pracipua Cofficarum qvantitatum pre- (7) Ewsdrra
prietas ex propofito problemate ejusdemqve altera folg- £tg"r ex
tione emergens eft, qvód idem inveniendus maximus ter- gU4NLHUUE-
"minus, nón fohim,ut antcà defionffratum, in qvantitatibus bir Cefficig
cofficis collectivé fumcis reperiatur, fed etiam qvod (inzu. EAR
Jis qvibuslibet pro ratione numeri terminorut dati inilt; Numer ue A
qvid?
numerus enlm terminorum plane hic idem.eft , ovóg in., terere
Cofficis e&penens diéitur ; Nah itiCafficis uaitatis cyoo- C" ^
pens eit c, Radicis /go) expanens veró r, qvadiati (q.) à, Co- t,
et cis (go) expanet Vero qvadiad (q.) ies usted.
^ T. .
3 Cojrcis,,
| e . CAP, FIL 00
Qvidft "bi (C^), Zenfi-ZenGi, (q.q.) 4. furdefolidi, (8.) y. Zenfi-Ca-
Fw] ^ bi, (qC.) 6.Bifurdcfolidi, (6.). 2. Zenfi-Zen(i-Zenfi, (qqq.)
sai .$. Cubi-Cubi, (CC.) 9. furdcfolidi Zeufi, (8q.) 1o. trifurde-
folidi, (tri) n. Zenfi-Zenli-Cubi (qqC.) n. qvadri-furdes
. folidi, (qradriB.) 1. Bifurdefolidi-Zenfi, (b9q.) 14. furde-
folidi Cubi (9C.) xy. Zenfi-Zenli-Zenfi-Zenfi, (qqqq.) 16. &c.
. Qyoniam itaqve in 6. 31. maximus terminus trium termi-
' porumerat proportionis datz Cubus unitate diminutus&
fic divifus per eandem proportionem unitate diminutam,
videmus, qvàd ipfius Cubi exponens etiam fit 3. Porró in.»
- $. 5. maximus terminus qvatuor terminorum erat propot-
tionis datz Zenfi-Zenfüs unitate diminutus, & per eandem
proportionem unitate dimirmutam divifus, Zen(i-Zeníus we-
. . rófimiliter habetexponentem 4. cum nümero termino-
| . rum coincidentem ; Deinde in, $.34. maximus terminus
qvinqveterminorum etat date proportionis furdefolidum
unitate diminutum & per eandem proporconem unitate,
diminutam divifüm , furdefolidi veró exponens, s. adhuc
coincidit cum numero terminorum, & tandem in$.5. ma»
ximus terminus erat proportionis date Zenli Cubus unita-
, tediminutus& per eandem proportionem unitate diminu-
tam divifüs, ubi rurfus Zen(i Cubi exponens, ó. idem e(t
cum numero terminorum ; Concludimus itaqve univería-
liter, numerum terminorum elle, exponentem Coíficum,
qvi exponit proportionis datz qvantitatem (Coilicam , qvas
Vuxe, Uuitatediminuta & ficdivifa. per ipfam proportionem da«
tePsinue — Cam unitate diminutam exhibet qvafitum progeffionis
gignetur — geometrice& maximum terminum , qvi ipfe etiam divifae
énculatim) per qvadratum proportionis datz , exhibet minimum ejus-
| € Purser "- demprogreffionis termjnumjitaqvidem;ut hi extremiser-
po "a minilecumunultiplicati producant fümmá omnium termi-
roamnus. iorum. | $53. Hifoe
Áh 202€ en d [^ tore t e d cc erf Lu f 24
Cc$pg «co?
. C45. PF . : 27
—— 78.38. Hifce przfüppofitis feqvicur, ex fieprus dicto pro- Tertiepro-—
blemate Clovii & noftra folatione; tertia. & mirabilis faneériere co/-
proprietas qvarritatum cofficarum,; qvód nimirum ezdem fea mimbi-
ab mátate collectieà five fammatim accepte, fint eqvales'
qvantitati cofficz inordine proxime feqventi,fed unitate,
diminutz, & per radicem unitate dimieutam divifz ; Ex gr.
fi fic proprefio ab "unitate geometrica in proportione,
Qvintupla & qvidem fex terminorum , continuó cum pro-
portione, 5. multiplicatorum, qvorum primus eft 1. fecun-
dus 5. ceu Radix tertius, »5. ceu qvadratum, qvartus 1:5. ceu
- ACCubus, qvintus 655. ceu Zenfizzenfus & (extus ji. ceu für-
defolidum ; Dico : horum'omnium fümmam (inveniendam
| per $39.) effe aqvalem Zentr Cubó 625, unitate diminuto
&& lic divifo per radicem qvantitátis cofüce unitate dimi-
&tatn, nempé 4. qvotiensenim 3906.eft fumma pradicts. DE
4.39. Cum veró per hosipfosexzponentes Cofficos in, Coskervm
boc & (imilibus cafibus tabulis noftra Tetr:zonometriz in- TP ene-
fignis praxis accrefcat , qvod ipfum vel foia qvantitatum minaziswi
«officarur verba infinuare poffunt, in graciam illornm,qvi ;,,. 0) e
XCotficie dodtring plane igharifünt, ex qvalesqvanti-Cufe,
tatum cofficarum fint figmificatinnes? qvomodo ulterids
procedendo feqventes vecentur? & qvisfitexponentium u-
fus? Referuntur veró omnes qvantitates Coificz per radicis
&ontinuammukiplicationem ordinecrelcentes , in univer-
fum & principsliter velad qvadratum numerum, five Zen-
fumvelad Cubutn numerum, vel ad furdefolidum; qvi ne-
ve qvadratus neqve Cubus eft ; Ex hifce tribus reliqvz
«vantitatesomnes& in infinitum usqve poffunt effe com-
polite, itaqvidem & perpetuis hislegibus, [1] ut ejusmodi
exponens, qvi exacte dividi poteft per numerum ternarium
(Cobus (it & qvidem illius qvantitatis, qvz habet exponen-
os P3 tem
d
qn é CAP, FIL
cem qvotientis. Ex. gr. cofficus exponens, 36. :potefh exacte
.- dividi per, qvarc, exponens 36. habebit qvantitatem coff-
cam,.qvz dicitur Cubus Zenzi-Zeníi-Cubi., vel ordine in-
versó, Zenfi-Zenfi-Cubi-Cubus, .qvia qvotiens n. eft expo-
. . nens Zenfi-Zenfi-Cubi ; Similiter exponens 9. exacté poteft
Denomin4, dividi per 3, qvare 9. eritexponens Cubi-Gubi, qvi qvoti-
8109914 Cofj- ( . . . .
c4, Lex (a) €a5. e(t exponens Cubj, [2] Ut ejusmodi axponens, qvi cx-
de quadrato actà dividi poteft per nymaerum biparium Zenfus fit, & qvi-
pti Zem[v. dem illius qvantitatis, qva habet: exponontem qvotientis ;
. Ex.gr. exponensto. exacté poteft, dividi per . qvare 1o. erit
, exponens zenfi furdefolidi, vel inversÓ otdipe dicendo, fur-
, de(olidi Zenfi, qvia qvotiens 5. eft exponens furdefolidi. sic
. exponens z;. eít Zenías Zenfi-Zenfi-Zeufi-Zen(i, propter
Denemina-. v otientem 16. [3] ut omnes exponentes , qvi per ternari--
"4 Le M Ea um & bjnarium dividi non poffunt, furdefolidorum nomi-
fov furdsfo- nc yeniant; & qvidem. ita, ut (i plan& per nullum numc-
lide. —. - rumdividi qveant, atqve fic fint primi numéri, (ecundum
()minume- definitionem Euclidis 1. VII, ifti exponentes dicantur fim-
yi primi, pliciter furdefolidigninores veró à majoribus diftingvantur
| f&candum ordinem eosdem numerando & nutmerum ordi-
nis preponendo «x. gr, quadri- fardeíolidus &c ;. (unt venó
. &xpoRentesinfrà centum omnes (urdefolidi feqyentes 5. 7.u.
Numeei pr V. 17. 19: 34-39. 35. 37. 41.43.47. 53- 59. 61. 67. 71.73: 79. 8389.97.
jin Pum QNérCC*. gr. exponens 41.-erit unde-fardefolidus, qviais
(y rernari- Jatdefolidorum eftundecimus, Similiter exponens ;7.cft bi-
gm infrá fürdefoljdus, qvia in ordine furdefolidorutm eft tbcnndus
gentum,qui?. &c, Si vero potfinr per aliqvem aliptp prater bihavium &
(2)In tume- ternarjum. numerum dividi à furdefolidis & diviforis &
pies 4b (s qvotientis fua (ortientur nomina ejusmodi expondntes. Ex.
Pm e rer gr. exponens a5. dici debet fardefolidi: furdefolideto, & ex-
pariym — ponens 3j. dicetur, bifurdofolidi (urdefolidum ; qvia fj divi-
egmpofitio, | doni
; ear bf * 0 7)
dàm 3s. pet éxpotientemi furdefolidi ;. Votietis erit 7, expo« -
&ens bifurdefolidi ; Idem cóncipe dé dliis omnibus. — '
5.30. Qyérnadmodum jam exponentes coffici ad déno- Pxposenrr-
fnifiandas qvimtitates cofficas infignem habent ufüfn , 1ia.. im 4d ne
ctiatn iidem exponentes& tabulis qvadratorum & tabulis jj, yt .
. Cuborüm ( qvas feré elaboratas poffideo & (fuis ufibus |
a1dornatus DEO danré, publico.donabo] haud fpernéndum
afferunt commodum j expohunt enim exponentés utrobi-
qve; qv& & qvomodo omnes qvantitates ex continua mul-
tiplicatione fen progreffione facta, intabulis inveniri qve-
ant & debeant, & ut taceam hic Cubicas tabulas, adufuün] ^
tabularum qvadratorum noftrornin fplüm | nos applicabie mU
mus, przmitterido univerfales exponehtium proprietátes, Expomewti
quarum prima eft, qvód qvilibet exponens cuilibet alii ad-
ditus, faciat exponentem, liabentem qvantitatem cofficam, falis prima
S^ huie o. 0 falie prima,
qvz per multiplicationem producitur. iftarum qvantita-
tum, qvarutn expohentesfunt additi. "^... ^ «5 «4
——— $. 4. Imprimis ergà intibülis quadratorum habetur Qu
quantitas exponentis 2, nempé ipfius Zenfi, five quadrati, qam HAM. -
quia radicis exponens, eidemesponenti additus Fatit 2. ex» 7, ponentis
ponentem Zenfi five quadrati, [ed & radix mültiplicata. a,
cum radice producit quadratum ; Deinde in tabulis habe- 6,
tur quantitas exponentis 4. quia in tabulis; quilibet qua- ez5openris
dratus minor quàm 160060. poteft affumi ceu radix cujus e. -
quadratum, erit Zenfi-Zenfus, five quadratiquadratum,ha- — ^
bens dictum exponentem 4. nenipé exponentem Zenfi fibi |
ipfi additum, alii veró Zenfi-Zenfi in tabulis non reperiun-
tur, quàm quorum radix non é(t major quàm 316. — Porró
in tabulis habetur quantitas expohentis $. ceu exponentis
4. fibi ipfi additi; quia in tabulis quilibet Zenli-Zenfus mi-
nor quàm roocoo, ceu radix ffumtus in iisdem fuum Ba- )
| ct
£45 HnIUCF-
Qvantitt
expeneuts
umproprie» —.
*
p—— pM -
[2 CAP. N.
bet quadratum, quod erit Zenfi-Zen(i- Zenfus, alii veró in
tabulis rion dantur, quàm quorum radix non eft major
. quam 17. quia radicis rg. quadratum eff, 324. & hujus qua-
. dratum, nimirum Zenfi-Zenfus radicis 1g. e(t 104976. cujus
adratum ih tabulis non datur, nifi id ipfum per $.3. c. 4-
quarere velis. Ulteriüs in tabulis datur quantitasexponen-
tis 16. ceu facti ex additione exponentis $. fecum ipfo, nimi-
rum Zenfi-Zenfi-Zenfi Zenfus, plures verà in tabulis non,
dantur, quam querum radix non eft major quàm 4, quia.
radicis . Zenfi-Zenfi-Zenfus, eít390625. cujus quadratum in
antira, tabulis rurfusnon datur ; Tandem exponentis 3z. quantitas
exponat. femel tantum in tabulis datur, quando nimirum radix eft z.
j^ hujus enim Zenífus eft 4. Zenfi-Zenfus 16. Zenfi-Zenfi-Zer»
| fus, 256. Zenfi-Zenfi-Zenfi-Zenfus, 65556. cujus quadratum in
tabulis, 4294. 967296. eft ipfe Zenfi-Zenfi-Zenfi-Zenfi-Zen-
fus. |
6.42. Omnesfic exponentes ab unitate in dupla pro-
We pregre^" portione inventi à; 1.2.4.8.16.32. 64. &c.. defignant
pne geome- portione Inventi, nempc, 1.2. 4. 8.1 31,04.&c. de ignant
trica expe. quantitates, qua per quadraturam continuam determinari
sess qvi? poffunt Quapropter fi in progreflione continua cujusli-
2 bet proportionis Pieu. denoioinstionis minimus terminus
concipiatur tanquam unitas habens exponentem o. deno-
minatio veró proportionis tanquam radix, erit numerus
uÁ terminorum unitate diminutus ipfe exponens.
» Pr eni- 6. 43. Hinc, fj detur denominatio proportionis, inpro-
me geomc- Bresfione ab unitate, & quinque terminorum, & quaratur
erica; f de- quintus terminus, erit is, Zenfi-Zenfüs date proportionis,
eur denomi- quia numerus terminorum unitate diminutus facit expo-
s4t10 P79-. yentem 4.. Ejg. quzritur quintus ab unitate terminus in,
M oemode proportione continua 235. eritq; 3049.800625. quia propor-
ivoemiaryr tionis data quadratum eft 55225. hujusque quadratum, qua-
quinti ter». fitus
SHINE, |
Svenisitar
exponentis
i6
- CA PR. y1L - Mh
' fitusqvintus terminus ; Similiter (i detur progreffto ab nni»
tate in proportione 12, duodecupla; Zeifi Zeníus numeri t,
hoc eft qvadrati144. qvadratum 2076. e(t qvafitus qvintus
terminus, 1 : DEN MEE
| $ 44. Si porrà detur proportio progreffionisab unita. Q*"ede
te & novem terminorum & qvzratur nonus tertninus ; erit len Piu
is propter exponentem 8. Zenfi-Zenfi-Zenfus proportjonis —.
datz. E. g. proportionis 1j. Zenfj - Zenfi- Zenfüs e(t --
2562.890625. eít qvxfitus nonus progreffionis terminus, qvia
| proportionis date qvadratum eft 225. & hujus. qvadratum,
$0625. & hujus qvadratum qvazf(itus nonus terminus. | |
S. 45. Si uerius detur proportio progreftionis ab uni- ueveZe
tate & 17. terminorum, & qvzratur decimus feptimus ter» 4tcimu fe
minus ; erit propter exponentem 16. Zen(i- Zenfi - Zen(j-7 rial .
Zenfus, proportionis datz qvaíitus decimus feptimus termi- |
nus ; Ex gr. date proportionis 3. qvafitus decimus feptimus |
terminus in progreffione ab unitate, eft 43.046711. qvia,
cjusdem qvadratum eft 9. & hujus qvadratum $t. & hujus
qyadratum 656r. & hujus qvadratum, qvafuus decimus fe»
ptimus terminus, ZEN v |
| 6. 46. Cüm in arithmetica jocofa foleat proponi ali- vermeds
qvis cafus in qvo progreffionis geometrice qvzritur trigefi- Prigefimus
mus fecundus vel etiam 35tius terminus, nolo hic fuübfiftere mmm P
. ininventione termini decimi feptimi; Proponi verd fóler ——
iJle cafus hóc modo ; Qvidam alieni generofi eqvi cupidus, Pro/lema
4 Domino hoc accepit refponfum ; Tibi lubentiffime ce- ex A4rizb-
damEqvum, fi pro calcei ipfius claviculis, numeró 22. & in- P€/ic4 Je»
fuperuna, in cafüm alicujus deperditz diligenter affervan- cof. HT
da, mihi (olvas rantám pecuniam, qvantam efficiet fumma, ,,, 75,
fj pro primaclavicula tantüm folvatur unicus nummus,. pro rs» cops-
- fecundayero folyantur duo nummi, pro tertia qvatuor, pro peediofiftoià
| M 00 T | qvarra/Piwm,
Áo
«
ceo 0700. o €AX3P FINO,
*
qvarta octo nummi & fic. porró continué dupló plus pro fía-
gulis ad ultimam usqve clavieulatuqvaritur jam, qvot hifi.
pro trige(irna tertia clavicula folvi debeant,ut poítéá tota.
nummorum fumma per $.:9.hujus capitis détermipari qve-
. "at, qvoniam itaqve numerus terminorum unitate diminu-
. -. tus eftexponens31. & proportio data z. ceu radix, erit iplius
"qvadtratum 4, & hujus qvadratum16, & hujus qvadratunt,
256. & hujus qvadracutn 65556. & tandetn hujus qvadratut
.00— 4194.967196. qvifitus trigefimus tertiüis terminus, qvia ab
exponente 2. ih dupla proportione usqve ad exponentem,
muim qvadratum 4. adhuc qvatuor qvadrata continià ya-
tione fümi debent ; Invertó fic ultimó termino, determina.
tio fümmz facilis eft per $.29. hujus capitis,erunt nempe --
$589934591. tummorum feu imperialium, 29. milliones cum
' éXlotonnis auri: 1 fallor atqve infüper $2616 r reliqvis
' groffis & numtnisex fingulari gratià remiffis, qvod prounó
XZ:nspre CO generofitfimo qvàmvis fatis iniqvum eft pretium, deo
areffene caetero absqve exponentibus & tabulie in vulgari arithmeti-
qgeametriea, ca omnes 3; termini ordine inveniri & poni debuilTent uni- :
4 quolibet. üs termini ultimi gratià, — |
Aur | cipidre E 47. Stifficiant premiffa de progreffione ab unitate;
ge, aeruy d, 1 VéTÓ primus progre(lionis terminus non fit ubitas; fed qvi-
sominario libetalius numerus datus, & detur (iml proportionis -de-
proporse- nominatió & qvxratur vel qvintus vel nonus vel decimus
fiey 400907 (eptimus vel trigefimus tertius &c, terminus ; adhuc etiam,
. 4lo qvintié. cut in precedentibus cafibus, qvadrata continu? qezrun-
4yel nonut "0 . ) A vo.
wel iymur, Ur» proportionis dat&pro ratione exponentium,defignáto-
"wel .,ja4 rum à nümero terminorum unitate diminuto, ultimum ve«
*e/inve». YÓ proportionis qvadratum multiplicandum eft'cum dato
"itur? primo termino, $ic ex. gr proportionis data 1j. Zenfi-
| E O7 -- | ' .— Xehfi.
E 31. inclu(ivé exiftunt qvatuor tertrini, qvare etiam poft pri«
- proportione dupla1o.20. &c. exponenti o. accedit 18. & fic
V -
Dc | C49. V1X ^ 07 7 HM
Zenfi-zeriftt 8i 73072 1. rnltiplicatus cum dato inima
^. termino 1000. producit 815730711000. qvafitum nonu ters
minum, Sicporró ex. gr. proportionis datz 117. Zenfi-Zetv
fus e(t 1217. 573921. cum dato minitmo progreffionis terminó £xempl. m
100, multiplicatus221777392100. eft qva(icus qvintus progres-
fionisarithmeticz terminus. — M
$- A8. Sic itaqve qvantitates exponentium ab unitate Qvosodo
in proportione dupla progredientium per tabulas cum fuo re//2vorum
u(u ofterifz funt, qvomodo & reliqve exponentium qvanti- 2^».
tates, per eosdem exponentes inveniantur, jam videndum; 1 ^
. Imprimis fi (ingulisinventis exponentibus in dupla propor- P oantitetés
tione 2. 4.8. 16. 32. &c, addas unitatem prodibunt exponen- per sbulue
16$ 3. &. 9. 17. 3 &c. qvapropter fi fingulas qvantitates ex- irverian-
' penentium in: dupla proportione ab unitate, multiplices ^** ?
cum radiee, hoc eft, cum quantitate éxponeritis r. habebis -2vorodo
qvántitatés exponentjum 3. 5. 9. 17. 33. &c. nimirum Cubum, PP !
furdefolidum. Cubi Cubum, qvini-fardefolidum, trifür-7,, 7; er;
defolidi Cubum &c, ü tu - |
6. 49. Hifce qvantitatibus fic inventis per hos invenien- ,QoomoZe
tur qvantitates exponentium , qvi jam inventis in pro- exponenti-
portione dupla accedunt ; Si& enim exponenti j/accedunt 6 10. 14
in proportione dupla 6.12. 24. &c, exponenti s. accedunt in dete
—-
porró 5 qvapropter (i (ingularum qvantitatam, qvarurm ex- |
ponentes funt 353:9.17 37. qvzrantur qvadrata &horem qva» * -
dratorum rurfus qvadrata & fic deinceps, emergent qvafitz
qvantitates exponentium 6. I2. 24. 10. 20. 18. &c, nimirum ^
Zenü-Cubus, Zenfi-Zenfi-Cubus, Zenfi-Zenfi-Zenfi-Cubus.
Surdc'olidi Zenfüs , Surdefolidi-Zenfi-Zen(íus, Zehi-Cubi- 755,,e ri
C€ybus; &c. - I -w9P 2. Us Mj.
$. 5o. Inventis fic qvantitatibus exponentium 6. Io. T2. 79.47.27. t,
. . | Qi " . 1$.
Y
(
-
t,
Exempl. IN
"-
Svomode —
HM 0l fub. FIL
13. 20. 24 &t. per hofce, fingulis his addendo ehitdtem ut
in $. 48. factum eft, prodibunt exponentes 7. n. 5j. Xy. at. 25.
&c. qvapropter fi fingulas qvantitates exponentium, 6. ro.
2.48. 20. 24. multiplices cum radice, qvantitate nimirum.
exponentis 1. habebis, qvantitates cxponentium , 7.11.1.19.
'" a3125.&c. nimirum Bifurdefolidum, trifürdefolidum, qva-
dri-fürdefolidut fefürdefolidum , Bifurdefolidi Cubum,
vendo furdefolidi-furdefolidum &c. | '
" exponenti- 6. $t, Rurfus inventisqvantitatibus exponentium, 7.
$m 14.23. S. 19. 21. 25. fi alii exponentes iisdem in proportione dupla,
. Wir. accedant, prout in S. 49» factum eft, infrà exponentem 32..
: accedent 14. 22. 28. & 26. exponentes nimirum qvantitatum
bifurdefolidi Zenfi, trifurdefolidi Zenfi, qvadri-(ardefolidi
Gon de Zenfi & Bifurdefolidi-Zenfi-Zenfi.
vo
| menti 5.54, Ruríus fic inventis qvantitatibus, qvarum expo-
sun 1j. 27. nentes funt 14. 22. 26. «& »8. inveniuntur .qvanfítates, im- —
$9» 2j, G'c, frà 1. exponentium nempé 15.23. 27. & 29. per additionem
folàüm exponentis r. & confeqventer per mukiplicationem.
. ipfarum earundem qvantitatum cun radice, eruntqve, fur-
defolidi Cubus, Septi-fürdefoliduin, Cubi-Cubi-Cubus& o- -
&i-fardefolidum.
Qevmo 6.53. Tandem reltant adhnc iofrà 5. exponentes jo. &
vaponemt* a. fedille invenitur per exponentem 15. in de
wm gon p. fedi per exponentem 15. in precedenti S. in-
6, ventum & hic porró per exponentem eundem inventum.
. .. go. ficutetiam perrà in exponentibus ultrà ga. progredi li-
«et, fi placeat.
Scbematx-— 8. 54. Qyomodo veró unus exponens ab aliis depende-
died at brevitatis & facilioris inventionis grati ex feqventi ap-
nirione &- v eet (chemate, in qvo fatis curiofas progreffiones e exami-
"want ab ad. DATO; bujus non eft loci. - !
gems : tx
o ME | | - 65.5
t
ll "CAP. VIL | Bg. 7—
-. 65,64. 6159.97. 59333-31949 47« 435* 43« 41 307 37354 33
64.62.6058. 56.544524)0«48«4.6.44..:42440.38. 364344320. ^
32. 31 30.292822 61254244 2:3« 224 21. 20. 19418 17. 16.
16. 30.1. 2.14426. 14244 D4 224 1I, 20.10, 184. 9.16, 8. —
':8: 114-144 7,14 DAI. 6. Il. 30. 10. $4 99 84 8. 4
«l4 7. 7. 6.324 66. 34. IO. $5 $4 8- 4-442.
2,-46.6.3 6.3. $$ 24 $ 44 44244-2214
6 12 42.2221. 422. 02. 1h
3221L2Lh40: 2141101 :b.
CURBMPB L L EMEN
p. : T 05. 4d probe.
*-— $.55. Qvoeniam fic emnes hactenüs qvantitates expo- »a Clav -
nentium infrà 32, imó etiam infrà 64. & porró (ic proceden- retre/w,
do omnium , per tabulas qvadratorum inveniri poffunt, e '
concluüdimus, problema à Clavio in $.30.propofitum, in tot M jA id :
progreffienis geometric cerminis folvi.poffe, qvot hacte- 44/4 jl
nus per tabulasinventz (unt qvaatitates cofficz, tüm bre- porejf.
vi(fimà tüm jücundiffimà vià ; Simül étiam qvamplurima.,
exempla jocofaad imitationem exenipli in $. 46. propofiti
proponi & unà eademqve folutione folvi pofle futficiat hic
veró unicum. Debent inveniri minimus & maximus, 24 -
terminorum continu proportionis triple, ita ut minimus pieinid |
terminus multriplicatus cum maximo, efficiat fummam o- ele
nium terminorum &ut maximus terminus fit fumma to- zs.
' tidem terminorum ejusdem proportionis triplz, cujus mi-
' nimus terminus eftunitas; Qvoniam itaqve juxta $. $. 36. &
37. duplici modo folvi poteft hoc problema, & qvidem ju-
xta $..36. per exponentem 2j. juxta $.37. veró per exponen- :
tem 24. ex fchemate veró $. 54. apparet, exponàntem 2j, (!) Jum
plures infrà fe habere exponentes, à qvibus. dependeat, $37
quàm expónensi4.adeüqve operationem perexponentem — .— —
| | 2j. pto-
"-— .
-
is
"
0N e . / 4
m 5 C4. FIL !
- 23. prolixioremfore, cómmodior folntio eft juxta $. 37. qva«
propter, cüm exponens 24. fub fe habeat. in fchemate expo-
nentes 12, 6. 3. 2. & 1. erit exponentis r. qvantitas, data pro-
portio 3. & qvantitas exponentis 2.qvadratum proportionis,
| 9. & qvantitas exponentis 3. Cubus proportionis, multipli-
cando qvadratum cum radice, fi tabula cubicz non (int ad
B manus, nimirum 27: & hujus qvadratum 729. répra(entabic
- ! exponentem 6. & porró qvadratum illius 5j1441. repra:fen- -
.' tabitexponentem rz, &tandem rurfusillius qvadratum, ju- X
|. — Xtag.j. C. 4. inveniendum, nenipé 2824. 295364.81. reprafen-
- « tátqvantitatem exponentis 24. qva unitate diminuta & fic
divifa per proportionem datam & unitate diminutatn, ni-
mirum 2. eft ipfe qva(itus maximus terminus, 141214768240.
minimus veró eft idem maximus, diviíns per qvantitatem.
^ | , exponentis 25. qv ftatim innotefcit, dividendo qvantita-
| D. tem inventam exponentis 24.. per radicem five proportio-
, - pem datam 3.erit nimirum qvotiens 9414.317882.7.
, (2) Ju*is, 8.56. Siveràfaltem probationis gratià velis eundem,
$3^ inaximumterminum invenire per exponentem, 23. juxta,
G. 36. invenies in fchemate fub exponente 25. cxponentes
22.IL IO. f» 44 24 I. qVapropter qvadratum proportionis
datz 9. reprefentabit exponentem 2, & illius qvadratum 8r.
exponentem 4. & qvadratum illud gt. multiplicatum cum,
' radice 5. nempé 245. reprafentabit exponentem . & illius ,
qvadratum 5904.9. reprafentabit exponentem.1o. atqve
. multiplicatum cum radice 3. hemp 177147, exponentem 11.
'" & illius qvadratum, juxta $. 3. cap. 4. inveniendum, ,
313. 81059609. reprz(entabit exponentem 22. atqve multi-
plicatum cum radice; 94143178827. exponentem ultimum
23.ceu ultimum terminum proportionis triple ab unitate. ,
24. terminorum, per qvem ultimum terminumjjuxta $. 2.9.
o; | EN mE hujus
- M.
€ 4 23 p1L EM ng
hoj jus capitis facilà invenitur fumma omniam terminoruttt
dividendo nimirum eundem inventü ultimum & primó tef-
minór, diminutum, per proportionem unitate diminutatn,
ut qvotiens fit, 4707158949- & eundem ultimü qyotienti ad»
. dendo, fumma enim 141214768240. eft, utanteà; qvaficus
maximus fecandum reqvifita fua terminus 4 Minimus qvz-
. fitus verü terminuseft antcà inventa qvantitasreprafentans
€xponentem 25. id e(t numerum terminorumunitate dimi- -
nutum,
$. 57. Fortaffis non difplicebit , pracedentem folutio- Idem exem -
nem applicare exemplo alicui jocofo, qvod in arithtneticis PI per
Libris paffim reperitur Aliqvis in fua Cella habet genero- nada
filfimi vini decem dolia, qvorum unum qvodqve continet datunt,.
"tentum amphoras, is fibi proponit aliqvem predivitem &
avárum decipere, qvapropter eidem offert mille iftas am-
phoras vini, fi bi & heredibus (his certó velit promittere,
. fpatió 24. annorutm, (ot enim qvodlibet doliorum habe«
bat Circulós ) parvos hauftusfecundum men(üram partis t
millefimz unius amphorz fingulis annis prebere &qvideà ^ —
- ta, uit primo annó tthicus hauftus, fecundó veró antó pro 5.
. pter multiplicatam forté familiam tres hauftus, tertió an^
'mó propter eandem rationem rutfus triplo plures, nimi- l
rum 27, hauftus & fic porró in feqventibus fingulis annis, Uo.
femper &aontinué triplo plures hauftus.prebeantur, usqve -
adultimutm annum 24. qvaritur jam, qvot in fumta haue
ftuspraberi debeant, & inventa eft anteà in S. $. preceden- '
tibus fumma 74/2/476$240, tot eim hauftus .praberi: de-
bent; jam veto, fiillam fummam dividas per mille hauftus -
habebis , 24421476? amphoras Vini; i.e: 141 milliones, & :
2/476? amphorarum, *& infuper 240 hauftus, qvos forté
avarus dare recufabit, ; "t
^ | | 6. 55. De :
" E 5 "E
Ve
"Altems ex-
!
»
^4
uM. BERE CAP. VIL.
penentium $55 De prima exponentium univerfali proprietate:
proprieta in qvantum ex additione corundem innoteíCere per multi-
eirca eerun. Plicationem & qvadraturam illorum qvantitates poffunt,
dem fubrm hactenus actum cft, feqvitur jam altera proprietas, qva eít,
ienem u-. qvód qvilibet exponens fübtratus ab alio relinqvat expo-
vim 4b «lio. pentem qvantitatis illius, qvz fit, fi majoris exponentis
qvantitas dividatur, per qvantitatem minoris fübtracli ex-.
ponentis , qvó etiam fit , ut dimidium alicujus exponentis
. denotet exponentem radicis qvadratz tertia pars verà
radicis cubice, qvinta pars radicis furde-folide &c.hactenus
qvidem plané adinftar Logarithmorum ; ficut ergó bene&-
" ció tabulae in $. 54. tradite imó absqve: illa à radice per
| multiplicationem & qvadraturam ad majores qvantitates
cofficas afcendere potuimus, fic etiam deícendendo ad ra.
dices per divifionem & radicis extractionem pervenire ,
easdemqve invenire poterimus, qvanqvam fatear, adhibi-
tis fimül tabulis cubicis multó compendiofius id peragi
Quomod» poffe, (it nihil ominuüs ad interim hzc regula pro folis ta-
pr" rbi bulis qvaaratorum ; qvód qvantitates exponentium nume-
Avtitz- ri imparis dividi debeant , perqvantitem proximé mino-
tes minores ris Cxponentis parium , alicujus pro lubitu affumtz radi-
per majores cis, atqve tuncjfi qvotiens fitidem cum affümta radice, vera
enis!- — radixa(fumta eft, (in minustàm diualia atqve alja affumen-
nr. : daeft, usqve düm qvotiens cum affümta | radice exafle can-
Banc 4; Bruno praxis verà hujus rei facilior & felicior eft, qvàám uc
prop o ab initio ante illam credi poflit ; U(us veró hujus rei in fe-
' malium ex. qventibus patebit. .
. gremüs imve- $. 59. 51 qvatuor continué proportionaliumy ecome-
um.
:
wire /tc9-. tricé dati (int extremi termini & qvaratür fecundus. termi»
termi-
pam, me- pus; tunc Radix Cubica ex qvadrato primi dati termini,
gieyum prj. ultiplicato cum altero extremo dato, eft qvafitus fecun-
| dus
v 000 $MSXKIR 0000. 0» |
dus terminusy Ralegamus hic: exempla adtábulis Cubicss, Kleid o-
cüm nullo labore ihidem folà evolptione expediri. queant; ^4eic-Cwe
$.60. si verü qvatuor continué proportionalium dati "^ ^ * .—
(inc extremi termini.& qvaeratur denominatio propprtio,
Dare 4. *
. e. |. . e. e. " 1
is; tunc radix cubica ek maximo termino per--raiponeda 2^ PVT |
diviio c&t-qvaGta denominatio proportionis; rurfus «exea rrepiziavi
pla tabulis cubicis committimus, qvamvis etiam per qvae sire drio- " .
* dratas elaborari poffint, ut-ex feqventibus colligere licet.
(x $ 6.Si denter qvinqve continué proportionalium ep. Prpertion,.
e
.»*
fremi terrent & qvaratut fecundus terminus,.tunc.radik " , —
Zeníi-Zenfica ex cubo primi dati termini multiplicato cum Dass quit.
altero extremo dato,eít qvafitus (ecundus terminus, ex; gr. ADM itm
Cübus primi dati, 9. nempé 79. multiplicatus cum altero exeremis in. '
£xtremo:. dato 864.5364 09.. producit Zenfíi» Zénfüm, , venire fe-
6302470411615 exponens Zen(i-Zen(i, eft 4. fub fe habens fu- ("4m rer
um dimidium a, qVare per extractionemy radicis qvadratz ,/772 4
juxta $. 1o. cap; 4. hic invenitur primó .Zeníus 79:881. cujus. primum,
sadix 891. eít qvafitus fecundusterminus, . —— C Eg Bá |
t es tt ew 20 Lo LL OEximiülio
6. 61. Sidentur qvinqve contiou?, .proportionaliutm. ,,4j; Zey.
extremi termini & qveratur denominatio . proportiomás 5 fi- Zenfice.
tunc radix Zenfi-Zeníica ex majore extrenro per. minus da- pa; avis.
tum extremum divifüm e(t qvefita proportio. Ex. gr. ma. sve prepor.
jus extremum 364516409. divifum per minus extremum 9. Pemelium
prodit qvorieaeem 9605960r. qvi eft.Zenf-Zenfus qvafitg 7 70m ie -
proportioiis, jam veró Zenfi-Zenfi exponens eft. 4. qvi pot mominsria
' dimidiationem fit z. & rurfus . per dimidiationem t. qvare, peor pruper.
. £a; erit illa 99, qvalita proportio.
fi éx 960596or. extrahatur radix qvadrata, erit illa ogor. & fi eism,
porró ex hoc numero, ceu Zenfo, extrabatur radix .qvadra-
$:63. Si dentur 6. continué proportienalium extremi Dar f
& qvaramy (ecundus terminus, tunc radix ferdeíolida ex préperrie-
-
0 i . 0 PTMBPDR, DM
C fgdiémdué- Zeníi ando primislali ccrthini; muktipficáts crim afterb
- dor extr9- cermino dato, eft qva(itus (ecundusterminus. Ex, gr: priu.
co ries PP terminidati 9. Zenf-Kenfus 6561. multiplicatus cutn altero
fum. : extremo dto, $735906i;. producit 5/66503906x;. fürdefoli-
ey mesia-. dunvqveüci fecundi térmiri, ergó radix furdefolida erit exe
qum pri-. . trahenda, qvod ip(ümqvomode fieri poflit, arithrnetici vix
WEN. audenttradere, qvàmvis tradi poffit, varius labor véró nihil
* ominus erit intolerabilis, càm econtrà tabule qvadratec
Kem rummirum immodgom radicem fuürdefolidam facilim$ ex- -
"nidicie fWr- hibeant illi , qvi-in tabulis illié fufficienter exercitatus eft
5: Érgvidem (éqvenusmin. modum; Extrihendu c(t radix (uc-
. . .'defolidaex prodi£to numero 476650390625. - Primó ommni-
S 0. maffumo ex tabulis qvadratum aliqvod magnum ex, gr.
. *, - wW9880j6ocoo.pro. divifore datifurdefolidi, ut folummodà
| ^'^ Uwideam, qvotnumerrin qvotiente fint futuri, qroniam igi-
"ur diviforis inicialis 9. major eft initiali dividendt;. & qvia
o5. .Mivifor habetio. numeros, reqvirentur in dividendo rr. gu-
. .. Hmeriprouno qvotiente faciendo, (ed qvoniam dividendus
a. 5, "emntüm babet daqdecim numeros, appayet exinde por:ejuss —
ow 0 Imedidivilienemvin qvotierite tanrüm dios numeros,qvales
os 05 t7 «qvalesetiam (int; qvod hactenus,non qvaricor, futuros effe
ci50c 0 jam verf diviferis radicem'9940o.accipias tanqvam qva-
^. "ratum, vides, qvéd ejuswcl-faltem prozimé: miseris qva».
DC ' dati numeri;99115: radix st. habeat: tres numeros, qvia» ^.
(. .Neró juxta $.5];qvotiens radici debeteffeeqvalis, fic veré
e hec ipsó majorcfiet, hinc aliudqvadratum pro divifore cft
. ^. eligendum;qvod in qvetiente faltem tot producat nume-
Aros, qvot habet diviforis rádixZenfi-Zenfica ; qvod ipfum;
» Werócveriet in qvadrato-habente-utanteà fo, numeros. d. |
3 woTCujessmitialisminor eft qvàm'dividendi initialis 2qvale o-
| ""muáüim, primwin. & maximum eft.qvadratum. $766417969. :
pe i
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per qvod ff dividemusfurdefolidun abiqve uliropefattóhé ^^
foló-initialium $nttitur habebitius qvotientem 196. jafri.
veró ejusdem qvadrati rádix, 7/937: fl tanqvara qvadvatum,
con(ideretur, erit ejüs vel falten. qvadrati proxime fmino-,
ris radix, 275. fed mult$ major qvam qvotiens roo. Affume, .
aliam minorem radicem e. g. 245. cujus qvadratum 60018
K cujus porró qvadratum eft 3665. &c. ha&tenus.epim: tani.
tüm initialibus qvadratorum opus eft, per qvos initiales vol
faltem primum eorum, debet explorari fürdefolidum da».
€um primi folüm qvotientisergó, ut videamnus;an ille «tiam,
ficut initialis radicis fit z (ed: ftatim apparetadhuc effe,1.qvià.
initialis qvadrati nempé 5. in initiali furdefolidi, $.tantàp — .
femel centinetur, initialis qvotientis, non erit 2. ni(iinitia-
les qvadrati faltem (int dimidium initialium furdefolidi ag» — .
:qvefic qvotiens mero intuitu 260. qvod ipfüm omniu pri--
-
-€ .
*
i
*
" «nó continget in qvadrato.cujus inkigles fünt 388324. &c. ca-
-jusqve radix 5695. tahqvam qvadratum habet radicem 25.
'& qvoniam qvotiens 300. adhuc minor cft qvàm radix 25.
-Sveraturergó. qvadratum minus priori,cujus initiales divj-
:- dunt initiales fürdefolidi, ut. fecundus qvotiens (ii 1, vel'à,
qvale ex.gr, eft qvadeatum 2510/94. dec. per qvodáà qvotieg-
ite proveniunt 225. fed radicis, 0200. ceu qvadrati vadix:eft. 4... Ácon,, -
($24. Qvótiens.verÓ., 319, e»gó inter 224. & 228. Vera Tit rà- jp) aro.
|. Mix nempéforté incidendo in qvafitam radicem furdefoli- pergpnali,.
( wn dysbus. .
-darn 225. - ME ! ide
. CXIKEMIII 10g
$. 64. Si dentur fex continné proportionalium duo ex- "EUM
'trerni & qvaratur. denominatio proportieniss tunc majus rore
extremum dividendum erit per minus, & erit qvotientis ra- preportio-
- dix furdefolida qva(ita proportio, five ejus. denominatio. »«, . .
Ex. gr. majore extremó 87890615. divisó per minus CXtTE- corps laps,
, "umgserit qvorieas 9765625. Ra Mans eX qvo exttà- lis
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reperta ams veldenotinatio. proportionis; tuhc : utroque 'Cafu 9
tundá vict,
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raum in tabulisqvadra?umplané non poteft effe divifor dari
fardefolidiqvia illud i0.hoc veró 7. tantüm babec numeros;
«qvazrendo aliqvod qvadratum 7. numeros habens & minus
dató furdefolido, erit illud omnium primum & toaximutma
865615. per accidens cum furdefolido idem; qvotiens. ye-
rà in ejusmodi divifione tannit erit unitas, cüm econtrà fi
qvadrati illius radix j125. tanqvam qvadratum accipiatur,
erit cjus vel faltem proximé. minoris qvadrati radix, 55.
. major qvàm qvotiensi. affume ergó qvadratum qvoddam,
. fex: nuineros tantüm. habens. & cujas initialis eft minor,
— qvàm initialis farde[alidi, eritqve illud omnium primum &
inaximum, 976144. qvod primó intuitu dividit (urdefol£-
' &um ita, ut qvotiens fit.1o. fed i(tius quadrati radix 988. fi .
Sanquam quadratamaccipiatur, eric ejus velíaltem proxi-
ine minoris quadrati 961, tadix jt.'major quàm quotieniro.
Videmus jam quód ihter io, & 31. (it vera radix, quarc que-
rendum eft quadratum , quod dividit furdefolidum ita, ut
| primus quotiens fit 2.quod invenies, (i queras aliquod, quod ,
. habet: initiales,dimidió furglefolidi iritialium aquales,nem-
-p; 488. &c. quale invenies circa:radices 699: vel 698. jam
"veró, (i has radices ceu quadrata accipias, & eatumdem vel:
faltem proxime minoris quadrati radicem quaras, erit ille,
, 6. adeaque vera inter 20. & 16.'& fic tandem per lévinfcu-
lam attentionemobf(ervabis, quód circà radicem 55. (it ve-
t« radix furdefolida., quemadmodum etiam ipfa illa eft
"cm Zenfi-Zenfus ejus 390625. dividat furdefolidum etiam
per 1...
& 65. Si dentur feptem continu proportionalium dup
"extremi &, qveratur vil fecandus terminus, ceu medius pri-
pus
*ub - *
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és confeftim (cripta de página ad paginata typis tmando, in-
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medii funt 462, & 462, quare horum fumma facit vertica--
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requirunt' radicig inyentx omnes .& fingulas quantitates
cofficas ab exponente propofito fed unitate diminuto usque
ad exporientem 1. defcendentes; multiplicatas cum quan-
.titatibus cefficis fecundi numeri. radicis per primam opera«
: fionem inveniendi, ab exponente t, usque ad exponentem
propofitum fed tnitate:diminutum a(cendentes, neciíon.,.
: multiplicatas cum £fiugulis in (chemate rriangulari nume- |
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*iohali vel fecundó terminó vel etia denominatione pro-^wmvel — 7
" portionis , qvibüs invengis facile jam eft invenire reliqvos pli, i
, medios omnes,faltem continu? duobus inveniendo tertium ,,,jisyur, |
- proportionalem, veletjam continué mukiplicando ante- ao incom. P
cedentem terminumcum denominatione proportjionisad-gambils, - ^ ^
.tóut[ibi conftet incomparabilistabularum tetragoniarum —. 5. ^ 5/7 077
*[üsin multiplicandis ÷ndis continuà zatione nü«.. . 77. ^
. ametis, 003? MMC E
Cos $34. Alia haud fpernenda problemata proponit. d.1, Svonod ^
... Clavius, qvorumi-feqventia adhuc tetragonometrie noftre podio e"
— funtuffinia, Regulà enim norit proportionalitatis geon - Pertienalet (0
o-Ctricé docet, qvomodo poffimus, ihvenir € trés numcros 3- arirbinerich,
; rithmetrice proportionales, ita, ut.bini habeant urtim me- babentes.
' "lium proportionalerm, intellige sumeró.clfabilem vel ra-istermedios
tionalet, juxta $.3. cap. 4. nimirum fumendo tres numerose o Ue
Qvoscutiqve qvorum fecundus primieftqvintuplum & ter." * SERE
tius ejusdem primi fepruplum , horam enim Éngulorum., 0
. yvadrata erunt in proportione arithmetica & fimul intet
primum & fecundum nec nón inter fecundum X tertium, !
| frechóninter primum & tertium cadet unus medius £ea-, In
" gmetricé proportionalis & non folum inter alla gvadrata,, |
fed etia horum «qvé-mulüplicia per tnultiphcarronem,
eórum, vel fubmultiplicia, id eft per eorandem qvamlibet
ivifionem ; ibi porró admiratione dignim judicat Cla»
— Nini, inventis ejusmodi tribus arithmericé ;proportionali-
| bus, qvadratum medii geometrici intet primum & fecune.— -
dum arichmeticum, deinde qvadratum medii arithmetic ^ ^. S
«& deüiqve qvadratum medii geometricielfe in groportio- |
Aneéatithmética, Dantüritaqve pertabulas noflras muta — ^^ |.
. Amilliacxemplorum, qvibus hzc doctrina Clavii ad exam en : 2
"us ^ ) Qeduci 2
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deduci poteft; Si enirh cujuslibet radicis et. ge. 176. affu-
11, erunt harum trium radicum qvadrataj0976.& 774409.
& 1517824. in proportione arithtpetica, qvia differentia anter
primum & fecundum 743424. eít eadem, qve sater fecun-
cm dum &tertium, Sed & multiplicándo primum arithmeti-.
cum 30076. cum fccando 774400, habebis nümerum.qva».
dracum 23987814400. cujus radix, 154880.eft illorum medius
proportionalis, five etiam productum ex multiplicatione ,
. | primi affümti 176, cüm qvintuplo ejusdem 880. prout recte.
, .onetClavius:' Similiter mnltiplicande fecundum arith-
metícum 774400. cuni tertio 117814. habebis rurfus nume.
rum qvadratuni 175402905600. cujus radix 1084160. (ive ct»
jam productum ex tnultiplicatione primi a(lumti 176. cum
uu feptuplo ejasdem netmpé n3. —Porró qvadratum medii
: ;nter primiim & (ecundü arithmeticum nempe 33987314400.
ai dinis fecundi arithmetici j9969:g6o000: nec non» -
t5. t9
dvia ner. primum & fecundum quadratum Diferentia e(t
$75707545600. qVx etiam e(t inter qvadratum fecundum &.
fertium; Si porróinventorum ab initio qvadratorum affu;
fnas iisdem zqvemultiplicia qvzlibet, multiplicando nimis
. Yum eadem fingulà per unum eundemq; numerum ; Vel et-
iam'i eadem dividas per aliqvos numeros ( poteris verà
femper per illos numeros eorumve qvadrata,gvi dividunt
rimut riumerumaffuttum, ) adhuc proportio exifter a-
rithtnetica ; -Ex. gr, 11, dividit affumtum 176, qvars etiam
dividet npn foldm ipfe ,. fed & ejus qvadratum , fingula jn-
yenta tria qvadrata propertionalitatis arithmetica,
$75
máínufa & divifa per datam proportionem pnitate diming-
| P €4ÀP. Fn NS
.,. S. 7f. Tfi regula decima proportianalitatis Geometriee Qosmog,
tttragonometriz qviddamn affine tradit Claviut, qvód nini- /rvenian-
rum poffint inveniri duo numeri inter qvos cadant qvotli- /sr 4o na-
bet medii geometricé proportioriales integri & non frati, 7^ prid
fi affnrnantur nimirum daüo qvilibet numeri propartienem dvorlibes Uc
datani habentes & uterqve eorum toties continué fecum i- medi pto. —
píó multiplicetur, qvet debent cadere medii proportiama- perrionales -
les numeri, nnó adhuc infüper addito, producta enigy. erung Artgeó sen
p 5. "n : nift, vel
Mi numeri, inter qvos cadunt tot medii. proportiones, plan) irn
Sjvot reqvirunturin data proportione ; Jam veró qvomo- jio nelos,
do benefició exponentium, breviffimé &facilime per cabn-
las qvadratorum fuprá fufficientisfimé traditum cft, :
- $.76. Concludimus randem qvidem-cum Clayio pre- Nomwsap -
"portionafítareri geometriram problemate : Numerum. qvemijbor
qvemeunqye datum diltribuere in qvotvis partes propor, 4&"i&re
SE : A | : Onperter.
tionales geometricé in data proportione , fed multó com gesmietrici
pendiofiüs & artificiofiüs idern probletna folvimusutfequi- 5; a1,
tnr; Numerus, qvi determinat,in qvot parsres datus numne- (/) per in-
. rus dividi debeat, efto exponens defignans qvantitatem co(- vesriomem - -
ficam darz propprtionis, qve qvantitas cosfica uuitate di- Pe* nrij-
tam,datum numerum dividit per partem primam feu mini-
mam, qvà inventá & reliqvz omnesfacilé inveniüntur, fj ni-
' mirum eadem pars minima continué & totiescum data pro»
portione mu]tiplicetur, iniqvot partes datus numerus divi- '
di debet: Hoc problema, qvomodo illi qvodin$. 31. propo- *..—
ficum eff affine fit & duplici f(olutione gaudeat; baud finede-. -.—
lectatione dijudicari poterit ex $.37. &38. atqve nt hoc eó.—
facilijis fieri poffit, proponimus excmplum ex&. ss. defüm-
tum: Sit datus numerus 14114768240, qvi dividensps eff Keeplem — —
jn 24. partescondnué proporriornles & qvidem jn yatione
«ipla;
,
né 0020 fAPVFPMH
. E tripla ; quapropter qvantitas. npmeri' 3. fub. expohánte r4.
| . eitibidem 132419536481. qv& unitate diminuta & fic divifa.
per proportionem datam & unicate .diminutam nimirum,
141314768:40. datum numerum, diyidit peri. qvzlitam
mempé primam & minimam partem ,, qvare i (ccunda pars
etit ,tertia 9. qvarta 27: &c. omnes veró tandem in nnam.
fummam collecte efficient datum numerum 141214768240.
(imei is- ^ $.77. Si verénon prituam feu minimam partem primó
Ventionem-: omnium (ed.aliam partem qvatmlibet vel intermediam vel
jartié ot^ maximam qveramgs;tunc adhue qvidem datus numerus di-
Jetliber in- Vidi debetemper &univerfaliter per divif deter-
ermtdia emp per diviforem jam deter
Ve etiam. minatum,fed datus.numerusnon aliter fic dividitur;nifi pri-
'— Warme. ds multiplicatus fit cum. qvantitate coffica proportionis
E "date qvz habeat exponentem unitate minorem qvàm nu-
Us meruseft,. qvi determinat qvota pars.fit omnium primó irn-
Exemplum... venienda ; Ex. gr. fi qvinta pars (i t invenienda, datus nume-
ED - rusmultiplicanduseritcum Zenfi-Zenfo proportionis data,
. qviaexponens4, deignat t Zenfi-Zenfum,. & h C de cxteris, re-
teutó-femper communi divifore..
le . $.78. Ad proportionem mufitam feu hármonicam pro-
. wreportione gredimur,in qva neglectis illis, qya per. limplicem: multipli-
Mufica, (cu. Cationem.& divifionem peragi & ex Clavio peti poffunt, ali-
baremonicae. qya íolüam cxaminabimys,, qvorum cum qvadratis numeris:
Defnitio fuum eft. commercium ;. Harmonica veró feumufica pro-
Buys .. portioe(t, qvando. tribus.numeris propofítis, diffcrentia in-
7 Svomedi ter primum.& fecundum numcrum fe haber i inea ratione
er eritüme 9d differentiam inter fecundum & tertium. in. qya. ratione
sica proper .Á€ haljet primusnumerus ad.tertium.. 22
soweRanr — $79. Ex Arithtnetica veró. proportione trium termi-
sre? barm- norum gignitres terminos harmonica proportionis, fi pri-
- iei iwmeri? cus. terininus. multiplicet. fecundum. & tertium. termi-
Coo Voto ut M : num,
"et E .
r^
Ld .
— —ÀMA mA——Á— 0. 2880 RÀ.
HUCABRLNM, OW
aum , nec on fecundus tertium terminum arithme-Comgendie-
ticz proportionis, ut ita ifta tria muktiplicationis producta e" »e-
fint tresnumeri harmonicé proportionales, docet :Clavius lae.
in Regula prima proportionalitatisharmonicz; —— -
'$. 8o. Sed nos absqve ulla.multiplicatione eosdem tres
harmonicos terminos conftituemus, fi intabulis qveramus
' 4medii aríthmeticz proportionis termini qvadratum & ab 07.
illo qvadrato fübtrahamusqvadratum differentiz,feu excef-
fus in ifta proportrone arithmetica & habebimus tunc medi-.
umterminum harmonicum, ábhoc medio terminofübtra- ^. ^".
he porrà qvadratum primi termini arithmetice propertie- 20
tiis & refidui dimidium, fi addatur eidem qvadrato primi .
termini, erit fümma primus terminus harmonicus, fi verói-
dem dimidium addatur dicto qvadrato medii arithmetici &
infüper eidem qvadrato adhuc qvadratum differentiz, etit
trium illorum fumma tertius proportionis harmonica ter-
tinus; 'E.g. Sint tres numeri arithmeticé proportionales;ge, e],
10661.10988. & 11515. qvorum differentia eft 7 ; qvadratum.
medii termini eft 120.736144. à qvo fübtractum differentie:
' qvadratum ro6919. relinqvit: medium terminum barmoni-
cum 120619115. fi porró ab hoc medio fübtrahatur qvadra.
tum primi termini t5. 656621. relinqvitur 6972194. cujus di-
midium 3486147. ft addatur dicto qvadrato primi termini,
. erit fümma 17143068. primus terminus harmonicus, idens
veró dimidium,fi addatur dicto qvadrate medii arithmetioi
& infuper qvadratum dicte differentia erit funma horum,
trium 14329220. tertius harmonious terminus, Similitor
€x tribus arithtueticis 2. j. 4. fiunt tres harmonici 6.8.15. ,
nam medii Qvadratum eft 9. & qvadrato diforántiz dimi- Exemp ü
nutum 8. medius 'harmonicus , qvadrato primi diminutus
«i 4. eujus dimidium 2, fi addatur qvadrato primi 4. habes
Ta primum
ie -. m X4 9. FAL
primum harmonicum 6, idem etiam dimidium , & addatur
icto qvadrato eedii 9. babebis. ri. &fi porró hutc addas
«vadratuth differentiz habebis i1. tettium barmonitunt, vi-
descenim, qvód di&ercntiaz. inter primum & fecundum fe-
habeat ad differentiam inter fecundum & tertium 4» ficut
fíchabet primus 6. ad tertium i. — B
emodo — . $. E. Pernumerum qvadratum, folvit. Clavius curio- — .
inveniantur liffimum problema in Regula VIL proportionalitatis bar-
| qvinquena- mpnicead Euclidem. qvod ica fe haber: Sint inveeiendi -
| eri saben. qvinqve numeri, in qvibus omnes tres proportionalitates.
3o exiflunt, ita, ut prirni tresnumcri babeant proportienalita-
porísonet, . . . . . -
- eritbmericí 1008 arithmeticam , relictó qucem primó, tres infeqventes *
geomeiri-. proportionalitatem geonietricata hi tres po(lremi prapor-
. eem G. mu-tionalitatem harmonicam ; efficietur autem hoc B confti-
| 4", ^ tuantur tres tctmini in proportionalitate geometrica cu»
juslibet ptoüportionis & reprefentabunc illi tres, qvintum,
tertium & primum terminum, conftitutó jam medió termi«
nó arithmeticó inter qvintum.& tertium,jam adfunt omoes
prater fecundum, qvierit qvadratum tertii cermini, divi-
fü per qvartum terminum ; & (i aliqva. fractio accelferit,,
probletnati veró facisieri debeat in. numeris integris , te-
ducendi funt omnes qvinqve termini ad eandem denomi-
gationem, qvà neglettà numeratorcs erunt qvafi qvinqve
numeri, Ex gr. int conftituti qvintus,tertius & primus ter-
* minus in proportione: geometrica tz. 6.& 3. inter 14, & Ó.
snedius arithmceticus eft 9. qvartus terminus, qvi dividit
qvadratum tertii 36, per 4. fecundum terminum, fünt erg .
hi qvirtqve numeri 1. 9. 6. 4. 5. illi qvi qvarnatur.-
Sivemedo $. gi. Ad Tetragonomettiam noftram omninó ctiam
. srvenientur trabi poteft, qvod Claviusproponit ín Regula VIII, Har-
QUIT 9^ anópicz preport:ad Eucl, Inveniendi (unt qvatuor npume-
fis
" ! Esemplnom,
LEE
| CAP PRO c 0 av
fi qvonim extremi babeant datam. proporaenem & qvi
cétifticuanc Hanmoniam maximnans, hoc eft, wt ipfi fint Geo-
| ftietrict proportiorales, qvamvis non continué fedfaktem ,, 5.
dicretà, unus autem tuediorum cbfh extremis fervet pro-
porionalitatemi atithmericam , alter veró harmionicam.
meri M
tuentes Dar-
PONAM
. Sohitio efttalis : Affumi debent duo numeri qvilibet ha».
befitesdatám proportionem , inter qvos debet coníticul. —
. féditis arichmeticé proportionalis (qvod facil? fit,. cii
fémper fit diunidia (amma extremorum) per/hos tres arith-
- "fnerité prepotrionales juxta $.79. qvarancur tres barmoni-
tà proportionales qvi erünt tres termini, prius, fecundus
Sic éx. irr. duo nnmerj in dupla propertione t, & z. hébenk
. tmedium aritimiteticé proportionalem 3 ur ita fint. per re«
: p!
dudtionem ad eandem denominationem, neglectá denomi-
tatione, tres arithmeticé proportionales 2.3. 4. qvia jam
. qvadratum medii eft g. etfi (übtrahas ab hoc qvadratum dit
.. ferehtkz 1. re(iduum erit medius harmonicé proportionalis,
Aqvo fi porró fübtrahas qvadratum primi tertnini 4. refi-
& evártos quafiti, certius.veró e(t medius arithmetic& proe
. portionalis iater primum terminum & qvartum invéntus,
Exemplum,
-w
duum erit 4. cujus dimidium z. fi addatur qvadrato primi.
- termini 4. fumma 6, erit primus Harmgnicus & (i idern di-
inidigm unà cum qvadrato differentie addatur diclo qva-
drato medii 4richtmetici füumma trium erít 11, tertius har--
anonicus terminus, hi veró tres termini funt. qvatuor qvz-
fitorum primus, fecundus & qvartus, tertius verà c(t medi-
us arithmeticus inter primum & qvarcum 12. nimirum 9.
qvare hi qvatuor numeri 6, 8. 9.12, conítitunnt harmonia
maximam juxta definitionem qvidem Clavii, .
CD T3; 0 $fÀwve -
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2 CAP. VIL^ »
Mujucap. —— $.9 Atqve ficin heccapite omnes tres proportigBa-
ti 69 totiu ]itates, arithmetica, geometrica & mufiea ad tetraponome-
eperit pilo triam noftram applicatzfuere, ita, ut haud difficile jam fit
gus cum cognofcere, qvàm late fe diffundat hzc doctt ina & ufus ta-
fümmaria .— A .
^d snxycy. bularum tetragonicarum in univerfam Mathe, qvz cir-
famfer ca proportiones numerorum & figurarum tota fere vería- -
Geemetriamtur, qvid autem posit circa figuras Geometricas Tetrago-
introdu- — s ometria noftra, ex ipfis fontibus optimé hauriri poterit,
dion. — a idelicetex Elementis(1.) Euclidis & qvidem Libri I. pro-
pofitione 47. totius Gcometriz bafi precipna, ob cujus
inventionem Pythagoras centum boves immolaffe creditur,
* ex Libri II. propofitione 2. cum omnibus feqq. ex L. IE.
propof. 36. ex Libri VL propofitionibus ui. 13.17.19. 20. 25.30.
ex Libri VII, propofitionibus, .8. 9. 10. Ir. 13. 14.. 16. 18. 20. 22.
$24. & 16. ex Libri IX. propofitionibus 14,8. 9,10. 18. ex Libri
X. propofirionibus, 9. n. 15. 18. 19. & omnibus reliqvis de
qvantitatibus potentià commenfürabilibus agefitibus: ex
Libri XIII. XIV. XV. XVI. propofitionibus feré omnibus.
(2) Apolloniide fectionibus conicis , & qvidem exLibri L
prop-1/.20.21.27. 33. 49. $0.31. f2. 57. 54.55.36. ex Librill. prop.
3. 19. 11.22. 27. €X Libri III. propof. 15. 77. 1$. 29. 20. 21.22.:33.24.
2j. 26. 27.28.29. 4$. $2. $4. $5. $6, — Singularia veró aliaet-
. $am qvàm plurima in. Geometria practica tradere & fi-
guris competentibus illuftrare propofui qvidem mihi fed
fata hactenüs inhibuerunt , f(ufficiat jam tetragenomes
. £riam tabulariam dempnítraffe & exercuife .in. qvàm,
/— plurimis, ut ad infinitum ejusdem ufum aditus
jam fit perfacilis, in Infinitis veró jamet
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MAY 311940
Uii. OF MICH,
LIBRARY
UNIVERSITY OF MICHIGAN
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